Answering a question of Erd\u0151s and Graham, we show that for each fixed positive rational number x the number of ways to write x as a sum of reciprocals of distinct positive integers each at most n is 2(cx + o(1))n for an explicit constant cx increasing with x.
", "doi": "10.19086/da.154329", "issn": "2397-3129", "publisher": "Alliance of Diamond Open Access Journals", "publication": "Discrete Analysis", "publication_date": "2025-12-19", "volume": "2025", "pages": "28" }, { "id": "authors:q6ry1-7x950", "collection": "authors", "collection_id": "q6ry1-7x950", "cite_using_url": "https://authors.library.caltech.edu/records/q6ry1-7x950", "type": "article", "title": "The Largest Subgraph Without A Forbidden Induced Subgraph", "author": [ { "family_name": "Fox", "given_name": "Jacob" }, { "family_name": "Nenadov", "given_name": "Rajko", "orcid": "0009-0003-3777-021X" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" } ], "abstract": "We initiate the systematic study of the following Turán-type question. Suppose Γ is a graph with n vertices such that the edge density between any pair of subsets of vertices of size at least t is at most 1 - c, for some t and c > 0. What is the largest number of edges in a subgraph G ⊆ Γ which does not contain a fixed graph H as an induced subgraph or, more generally, which belongs to a hereditary property P? This provides a common generalization of two recently studied cases, namely Γ being a (pseudo-)random graph and a graph without a large complete bipartite subgraph. We focus on the interesting case where H is a bipartite graph. We determine the answer up to a constant factor with respect to n and t, for certain bipartite H and for Γ either a dense random graph or a Paley graph with a square number of vertices. In particular, our bounds match if H is a tree, or if one part of H has d vertices complete to the other part, all other vertices in that part have degree at most d, and the other part has sufficiently many vertices. As applications of the latter result, we answer a question of Alon, Krivelevich, and Samotij on the largest subgraph with a hereditary property which misses a bipartite graph, and determine up to a constant factor the largest number of edges in a string subgraph of Γ. The proofs are based on a variant of the dependent random choice and a novel approach for finding induced copies by inductively defining probability distributions supported on induced copies of smaller subgraphs.
", "doi": "10.1007/s00493-025-00190-y", "issn": "0209-9683", "publisher": "Springer Science and Business Media LLC", "publication": "Combinatorica", "publication_date": "2025-12", "series_number": "6", "volume": "45", "issue": "6", "pages": "60" }, { "id": "authors:9j8v9-dvk17", "collection": "authors", "collection_id": "9j8v9-dvk17", "cite_using_url": "https://authors.library.caltech.edu/records/9j8v9-dvk17", "type": "article", "title": "Sunflowers in Set Systems with Small VC-Dimension", "author": [ { "family_name": "Balogh", "given_name": "J\u00f3zsef" }, { "family_name": "Bernshteyn", "given_name": "Anton", "orcid": "0000-0001-9573-5357" }, { "family_name": "Delcourt", "given_name": "Michelle" }, { "family_name": "Ferber", "given_name": "Asaf", "orcid": "0000-0002-0568-4523" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" } ], "abstract": "A family of r distinct sets {A\u2081,...,Ar} is an r-sunflower if for all 1 \u2a7d i < j \u2a7d r and 1 \u2a7d i' < j' \u2a7d r, we have Ai ∩ Aj = Ai' ∩ Aj'. Erd\u0151s and Rado conjectured in 1960 that every family H of \u2113-element sets of size at least K(r)\u2113 contains an r-sunflower, where K(r) is some function that depends only on r. We prove that if H is a family of \u2113-element sets of VC-dimension at most d and |H| > (Cr(log d + log*\u2113))\u2113 for some absolute constant C > 0, then H contains an r-sunflower. This improves a recent result of Fox, Pach, and Suk. When d = 1, we obtain a sharp bound, namely that |H| > (r -1)\u2113 is sufficient. Along the way, we establish a strengthening of the Kahn–Kalai conjecture for set families of bounded VC-dimension, which is of independent interest.
", "doi": "10.1007/s00493-025-00186-8", "issn": "0209-9683", "publisher": "Springer Science and Business Media LLC", "publication": "Combinatorica", "publication_date": "2025-12", "series_number": "6", "volume": "45", "issue": "6", "pages": "58" }, { "id": "authors:xtea2-gmn29", "collection": "authors", "collection_id": "xtea2-gmn29", "cite_using_url": "https://authors.library.caltech.edu/records/xtea2-gmn29", "type": "article", "title": "Sharp Bound for the Erd\u0151s\u2013Straus Non-averaging Set Problem", "author": [ { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Zakharov", "given_name": "Dmitrii" } ], "abstract": "A set of integers A is non-averaging if there is no element a in A which can be written as an average of a subset of A not containing a. We show that the largest non-averaging subset of {1,…,n} has size n^(1/4+o(1)), thus solving the Erd\u0151s–Straus problem. We also determine the largest size of a non-averaging set in a d-dimensional box for any fixed d. Our main tool includes the structure theorem for the set of subset sums due to Conlon, Fox and the first author, together with a result about the structure of a point set in nearly convex position.
", "doi": "10.1007/s00039-025-00728-8", "issn": "1016-443X", "publisher": "Springer Science and Business Media LLC", "publication": "Geometric and Functional Analysis", "publication_date": "2025-12", "series_number": "6", "volume": "35", "issue": "6", "pages": "1712-1738" }, { "id": "authors:9mpsw-88x77", "collection": "authors", "collection_id": "9mpsw-88x77", "cite_using_url": "https://authors.library.caltech.edu/records/9mpsw-88x77", "type": "article", "title": "An explicit economical additive basis", "author": [ { "family_name": "Jain", "given_name": "Vishesh", "orcid": "0000-0002-7275-3218" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Sawhney", "given_name": "Mehtaab" }, { "family_name": "Zakharov", "given_name": "Dmitrii" } ], "abstract": "We present an explicit subset A ⊆ N = {0,1,…} such that A + A = N and for all ε >0,
\nWe present a short and simple proof of the celebrated hypergraph container theorem of Balogh–Morris–Samotij and Saxton–Thomason. On a high level, our argument utilises the idea of iteratively taking vertices of largest degree from an independent set and constructing a hypergraph of lower uniformity which preserves independent sets and inherits edge distribution. The original algorithms for constructing containers also remove in each step vertices of high degree, which are not in the independent set. Our modified algorithm postpones this until the end, which surprisingly results in a significantly simplified analysis.
", "doi": "10.1017/s0963548325000112", "issn": "0963-5483", "publisher": "Cambridge University Press (CUP)", "publication": "Combinatorics, Probability and Computing", "publication_date": "2025-05-16", "series_number": "5", "volume": "34", "issue": "5", "pages": "621-624" }, { "id": "authors:eebg4-rbt63", "collection": "authors", "collection_id": "eebg4-rbt63", "cite_using_url": "https://authors.library.caltech.edu/records/eebg4-rbt63", "type": "article", "title": "Set-Coloring Ramsey Numbers and Error-Correcting Codes Near the Zero-Rate Threshold", "author": [ { "family_name": "Conlon", "given_name": "David", "orcid": "0000-0001-5899-1829", "clpid": "Conlon-David" }, { "family_name": "Fox", "given_name": "Jacob", "orcid": "0000-0002-0664-497X" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Zhao", "given_name": "Yufei", "orcid": "0000-0002-1995-3755" } ], "abstract": "For positive integers n,r,s with r > s , the set-coloring Ramsey number R(n;r,s) is the minimum N such that if every edge of the complete graph KN receives a set of s colors from a palette of r colors, then there is a subset of n vertices where all of the edges between them receive a common color. If n is fixed and sr is less than and bounded away from 1−1n−1 , then R(n;r,s) is known to grow exponentially in r , while if sr is greater than and bounded away from 1−1n−1 , then R(n;r,s) is bounded. Here we prove bounds for R(n;r,s) in the intermediate range where sr is close to 1−1n−1 by establishing a connection to the maximum size of error-correcting codes near the zero-rate threshold.
", "doi": "10.1109/tit.2023.3327846", "issn": "0018-9448", "publisher": "IEEE", "publication": "IEEE Transactions on Information Theory", "publication_date": "2024-06", "series_number": "6", "volume": "70", "issue": "6", "pages": "4074-4078" }, { "id": "authors:sc4y5-c1068", "collection": "authors", "collection_id": "sc4y5-c1068", "cite_using_url": "https://authors.library.caltech.edu/records/sc4y5-c1068", "type": "article", "title": "On a conjecture of Talagrand on selector processes and a consequence on positive empirical processes", "author": [ { "family_name": "Park", "given_name": "Jinyoung", "orcid": "0000-0003-3962-5668" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" } ], "abstract": "For appropriate Gaussian processes, as a corollary of the majorizing measure theorem, Michel Talagrand (1987) proved that the event that the supremum is significantly larger than its expectation can be covered by a set of half-spaces whose sum of measures is small. We prove a conjecture of Talagrand that is the analog of this result in the Bernoulli-p setting, and answer a question of Talagrand on the analogous result for general positive empirical processes.
", "doi": "10.4007/annals.2024.199.3.6", "issn": "0003-486X", "publisher": "Annals of Mathematics", "publication": "Annals of Mathematics", "publication_date": "2024-05-01", "series_number": "3", "volume": "199", "issue": "3" }, { "id": "authors:zaber-9mz98", "collection": "authors", "collection_id": "zaber-9mz98", "cite_using_url": "https://authors.library.caltech.edu/records/zaber-9mz98", "type": "article", "title": "Regularity method and large deviation principles for the Erd\u0151s\u2013R\u00e9nyi hypergraph", "author": [ { "family_name": "Cook", "given_name": "Nicholas A.", "orcid": "0000-0002-1148-1642" }, { "family_name": "Dembo", "given_name": "Amir" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" } ], "abstract": "We develop a quantitative large deviations theory for random hypergraphs, which rests on tensor decomposition and counting lemmas under a novel family of cut-type norms. As our main application, we obtain sharp asymptotics for joint upper and lower tails of homomorphism counts in the r-uniform Erd\u0151s–Rényi hypergraph for any fixed r≥2, generalizing and improving on previous results for the Erd\u0151s–Rényi graph (r=2). The theory is sufficiently quantitative to allow the density of the hypergraph to vanish at a polynomial rate, and additionally yields tail asymptotics for other nonlinear functionals, such as induced homomorphism counts.
", "doi": "10.1215/00127094-2023-0029", "issn": "0012-7094", "publisher": "Duke University Press", "publication": "Duke Mathematical Journal", "publication_date": "2024-04-01", "series_number": "5", "volume": "173", "issue": "5" }, { "id": "authors:amyc1-hkk39", "collection": "authors", "collection_id": "amyc1-hkk39", "cite_using_url": "https://authors.library.caltech.edu/records/amyc1-hkk39", "type": "article", "title": "Small subsets with large sumset: Beyond the Cauchy\u2013Davenport bound", "author": [ { "family_name": "Fox", "given_name": "Jacob" }, { "family_name": "Luo", "given_name": "Sammy" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Zhou", "given_name": "Yunkun" } ], "abstract": "AbstractFor a subset \n$A$\n of an abelian group \n$G$\n, given its size \n$|A|$\n, its doubling \n$\\kappa =|A+A|/|A|$\n, and a parameter \n$s$\n which is small compared to \n$|A|$\n, we study the size of the largest sumset \n$A+A'$\n that can be guaranteed for a subset \n$A'$\n of \n$A$\n of size at most \n$s$\n. We show that a subset \n$A'\\subseteq A$\n of size at most \n$s$\n can be found so that \n$|A+A'| = \\Omega (\\!\\min\\! (\\kappa ^{1/3},s)|A|)$\n. Thus, a sumset significantly larger than the Cauchy\u2013Davenport bound can be guaranteed by a bounded size subset assuming that the doubling \n$\\kappa$\n is large. Building up on the same ideas, we resolve a conjecture of Bollob\u00e1s, Leader and Tiba that for subsets \n$A,B$\n of \n$\\mathbb{F}_p$\n of size at most \n$\\alpha p$\n for an appropriate constant \n$\\alpha \\gt 0$\n, one only needs three elements \n$b_1,b_2,b_3\\in B$\n to guarantee \n$|A+\\{b_1,b_2,b_3\\}|\\ge |A|+|B|-1$\n. Allowing the use of larger subsets \n$A'$\n, we show that for sets \n$A$\n of bounded doubling, one only needs a subset \n$A'$\n with \n$o(|A|)$\n elements to guarantee that \n$A+A'=A+A$\n. We also address another conjecture and a question raised by Bollob\u00e1s, Leader and Tiba on high-dimensional analogues and sets whose sumset cannot be saturated by a bounded size subset.", "doi": "10.1017/s0963548324000014", "issn": "0963-5483", "publisher": "Cambridge University Press (CUP)", "publication": "Combinatorics, Probability and Computing", "publication_date": "2024-02-21", "series_number": "4", "volume": "33", "issue": "4", "pages": "411-431" }, { "id": "authors:4x56w-qmn68", "collection": "authors", "collection_id": "4x56w-qmn68", "cite_using_url": "https://authors.library.caltech.edu/records/4x56w-qmn68", "type": "book_section", "title": "Universality of Spectral Independence with Applications to Fast Mixing in Spin Glasses", "author": [ { "family_name": "Anari", "given_name": "Nima" }, { "family_name": "Jain", "given_name": "Vishesh" }, { "family_name": "Koehler", "given_name": "Frederic" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Vuong", "given_name": "Thuy-Duong" } ], "abstract": "Let G be a d\u2010regular graph on n vertices. Frieze, Gould, Karo\u0144ski, and Pfender began the study of the following random spanning subgraph model H = H(G). Assign independently to each vertex v of G a uniform random number x (v) ∈ [0.1], and an edge (u, v) of G is an edge of H if and only if x(U) + x(v) ≥ 1. Addressing a problem of Alon and Wei, we prove that if d = o(n/(log n)¹²), then with high probability, for each nonnegative integer k ≤ d, there are (1 + o(1))n/(d + 1) vertices of degree k in H.
", "doi": "10.1002/rsa.21204", "issn": "1042-9832", "publisher": "Wiley", "publication": "Random Structures & Algorithms", "publication_date": "2023-11-28", "series_number": "4", "volume": "64", "issue": "4", "pages": "899-917" }, { "id": "authors:nkkgf-d2b28", "collection": "authors", "collection_id": "nkkgf-d2b28", "cite_using_url": "https://authors.library.caltech.edu/records/nkkgf-d2b28", "type": "conference_item", "title": "Optimal mixing of the down-up walk on independent sets of a given size", "author": [ { "family_name": "Jain", "given_name": "Vishesh", "orcid": "0000-0002-7275-3218" }, { "family_name": "Michelen", "given_name": "Marcus", "orcid": "0000-0002-7262-6892" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Vuong", "given_name": "Thuy-Duong", "orcid": "0000-0003-0271-9687" } ], "abstract": "Let G be a graph on n vertices of maximum degree $\\Delta$. We show that, for any $\\delta\\gt0$, the down-up walk on independent sets of size $k \\leq(1-\\delta) \\alpha_{c}(\\Delta) n$ mixes in time $O_{\\Delta, \\delta}(k \\log n)$, thereby resolving a conjecture of Davies and Perkins in an optimal form. Here, $\\alpha_{c}(\\Delta) n$ is the NP-hardness threshold for the problem of counting independent sets of a given size in a graph on n vertices of maximum degree $\\Delta$. Our mixing time has optimal dependence on $k, n$ for the entire range of k; previously, even polynomial mixing was not known. In fact, for $k=\\Omega_{\\Delta}(n)$ in this range, we establish a log-Sobolev inequality with optimal constant $\\Omega_{\\Delta, \\delta}(1 / n)$. At the heart of our proof are three new ingredients, which may be of independent interest. The first is a method for lifting $\\ell_{\\infty}$-independence from a suitable distribution on the discrete cube\u2014in this case, the hard-core model\u2014to the slice by proving stability of an Edgeworth expansion using a multivariate zero-free region for the base distribution. The second is a generalization of the Lee-Yau induction to prove log-Sobolev inequalities for distributions on the slice with considerably less symmetry than the uniform distribution. The third is a sharp decomposition-type result which provides a lossless comparison between the Dirichlet form of the original Markov chain and that of the so-called projected chain in the presence of a contractive coupling.", "doi": "10.1109/focs57990.2023.00101", "publisher": "IEEE", "publication": "2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)", "publication_date": "2023-11-06", "pages": "1665-1681" }, { "id": "authors:vkhej-47e23", "collection": "authors", "collection_id": "vkhej-47e23", "cite_using_url": "https://authors.library.caltech.edu/records/vkhej-47e23", "type": "article", "title": "A rigorous framework for the mean field limit of multilayer neural networks", "author": [ { "family_name": "Nguyen", "given_name": "Phan-Minh" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" } ], "abstract": "We develop a mathematically rigorous framework for multilayer neural networks in the mean field regime. As the network's widths increase, the network's learning trajectory is shown to be well captured by a meaningful and dynamically nonlinear limit (the\n mean field\n limit), which is characterized by a system of ODEs. Our framework applies to a broad range of network architectures, learning dynamics and network initializations. Central to the framework is the new idea of a\n neuronal embedding\n , which comprises of a non-evolving probability space that allows to embed neural networks of arbitrary widths. Using our framework, we prove several properties of large-width multilayer neural networks. Firstly we show that independent and identically distributed initializations cause strong degeneracy effects on the network's learning trajectory when the network's depth is at least four. Secondly we obtain several global convergence guarantees for feedforward multilayer networks under a number of different setups. These include two-layer and three-layer networks with independent and identically distributed initializations, and multilayer networks of arbitrary depths with a special type of correlated initializations that is motivated by the new concept of\n bidirectional diversity\n . Unlike previous works that rely on convexity, our results admit non-convex losses and hinge on a certain universal approximation property, which is a distinctive feature of infinite-width neural networks and it is shown to hold throughout the training process. Aside from being the first known results for global convergence of multilayer networks in the mean field regime, they demonstrate flexibility of our framework and incorporate several new ideas and insights that depart from the conventional convex optimization wisdom.", "doi": "10.4171/msl/42", "issn": "2520-2316", "publisher": "European Mathematical Society - EMS - Publishing House GmbH", "publication": "Mathematical Statistics and Learning", "publication_date": "2023-10-09", "series_number": "3", "volume": "6", "issue": "3", "pages": "201-357" }, { "id": "authors:2d31p-cjn16", "collection": "authors", "collection_id": "2d31p-cjn16", "cite_using_url": "https://authors.library.caltech.edu/records/2d31p-cjn16", "type": "article", "title": "A proof of the Kahn\u2013Kalai conjecture", "author": [ { "family_name": "Park", "given_name": "Jinyoung", "orcid": "0000-0003-3962-5668" }, { "family_name": "Pham", "given_name": "Huy", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" } ], "abstract": "Proving the \"expectation-threshold\" conjecture of Kahn and Kalai [Combin. Probab. Comput. 16 (2007), pp.\u00a0495\u2013502], we show that for any increasing property \n\n \n \n F\n \n \\mathcal {F}\n \n\n on a finite set \n\n \n X\n X\n \n\n, \n\\[\n\n \n \n \n p\n c\n \n (\n \n F\n \n )\n =\n O\n (\n q\n (\n \n F\n \n )\n log\n \u2061\n \u2113\n (\n \n F\n \n )\n )\n ,\n \n p_c(\\mathcal {F})=O(q(\\mathcal {F})\\log \\ell (\\mathcal {F})),\n \n\n\\]\n where \n\n \n \n \n p\n c\n \n (\n \n F\n \n )\n \n p_c(\\mathcal {F})\n \n\n and \n\n \n \n q\n (\n \n F\n \n )\n \n q(\\mathcal {F})\n \n\n are the threshold and \"expectation threshold\" of \n\n \n \n F\n \n \\mathcal {F}\n \n\n, and \n\n \n \n \u2113\n (\n \n F\n \n )\n \n \\ell (\\mathcal {F})\n \n\n is the maximum of \n\n \n 2\n 2\n \n\n and the maximum size of a minimal member of \n\n \n \n F\n \n \\mathcal {F}\n \n\n.
", "doi": "10.1090/jams/1028", "issn": "0894-0347", "publisher": "American Mathematical Society (AMS)", "publication": "Journal of the American Mathematical Society", "publication_date": "2023-08-07", "series_number": "1", "volume": "37", "issue": "1", "pages": "235-243" }, { "id": "authors:xfvsq-pqf42", "collection": "authors", "collection_id": "xfvsq-pqf42", "cite_using_url": "https://authors.library.caltech.edu/records/xfvsq-pqf42", "type": "article", "title": "Dimension reduction for maximum matchings and the Fastest Mixing Markov Chain", "author": [ { "family_name": "Jain", "given_name": "Vishesh", "orcid": "0000-0002-7275-3218" }, { "family_name": "Pham", "given_name": "Huy", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Vuong", "given_name": "Thuy-Duong", "orcid": "0000-0003-0271-9687" } ], "abstract": "Let\n \n \n G\n =\n (\n V\n ,\n E\n )\n \n \n be an undirected graph with maximum degree\n \n \u0394\n \n and vertex conductance\n \n \n \n \u03a8\n *\n \n \n (\n G\n )\n \n \n \n . We show that there exists a symmetric, stochastic matrix\n \n P\n \n , with off-diagonal entries supported on\n \n E\n \n , whose spectral gap\n \n \n \n \u03b3\n *\n \n \n (\n P\n )\n \n \n \n satisfies\n \n \n \n \n \n \u03a8\n *\n \n \n \n (\n G\n )\n \n 2\n \n /\n log\n \u0394\n \u2272\n \n \u03b3\n *\n \n \n (\n P\n )\n \n \u2272\n \n \u03a8\n *\n \n \n (\n G\n )\n \n .\n \n \n \n \n Our bound is optimal under the Small Set Expansion Hypothesis, and answers a question of Olesker-Taylor and Zanetti, who obtained such a result with\n \n \n log\n \u0394\n \n \n replaced by\n \n \n log\n |\n V\n |\n \n \n .\n \n \n In order to obtain our result, we show how to embed a negative-type semi-metric\n \n d\n \n defined on\n \n V\n \n into a negative-type semi-metric\n \n \n d\n \u2032\n \n \n supported in\n \n \n \u211d\n \n O\n (\n log\n \u0394\n )\n \n \n \n , such that the (fractional) matching number of the weighted graph\n \n \n (\n V\n ,\n E\n ,\n d\n )\n \n \n is approximately equal to that of\n \n \n (\n V\n ,\n E\n ,\n \n d\n \u2032\n \n )\n \n \n .", "doi": "10.5802/crmath.447", "issn": "1631-073X", "publisher": "MathDoc/Centre Mersenne", "publication": "Comptes Rendus. Math\u00e9matique", "publication_date": "2023-07-18", "series_number": "G5", "volume": "361", "issue": "G5", "pages": "869-876" }, { "id": "authors:ybh4f-j2f14", "collection": "authors", "collection_id": "ybh4f-j2f14", "cite_using_url": "https://authors.library.caltech.edu/records/ybh4f-j2f14", "type": "monograph", "title": "Homogeneous structures in subset sums and non-averaging sets", "author": [ { "family_name": "Conlon", "given_name": "David" }, { "family_name": "Fox", "given_name": "Jacob" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" } ], "abstract": "We show that for every positive integer $k$ there are positive constants $C$ and $c$ such that if $A$ is a subset of $\\{1, 2, \\dots, n\\}$ of size at least $C n^{1/k}$, then, for some $d \\leq k-1$, the set of subset sums of $A$ contains a homogeneous $d$-dimensional generalized arithmetic progression of size at least $c|A|^{d+1}$. This strengthens a result of Szemer\u00e9di and Vu, who proved a similar statement without the homogeneity condition. As an application, we make progress on the Erd\u0151s--Straus non-averaging sets problem, showing that every subset $A$ of $\\{1, 2, \\dots, n\\}$ of size at least $n^{\\sqrt{2} - 1 + o(1)}$ contains an element which is the average of two or more other elements of $A$. This gives the first polynomial improvement on a result of Erd\u0151s and S\u00e1rk\u00f6zy from 1990.", "doi": "10.48550/arxiv.2311.01416", "publisher": "arXiv", "publication_date": "2023" }, { "id": "authors:3bzbc-xj302", "collection": "authors", "collection_id": "3bzbc-xj302", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20221017-10817000.4", "type": "article", "title": "The upper logarithmic density of monochromatic subset sums", "author": [ { "family_name": "Conlon", "given_name": "David", "orcid": "0000-0001-5899-1829", "clpid": "Conlon-David" }, { "family_name": "Fox", "given_name": "Jacob", "clpid": "Fox-Jacob" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" } ], "abstract": "We show that in any two-coloring of the positive integers there is a color for which the set of positive integers that can be represented as a sum of distinct elements with this color has upper logarithmic density at least (2 + \u221a3)/4 and this is best possible. This answers a 40-year-old question of Erd\u0151s.", "doi": "10.1112/mtk.12167", "issn": "0025-5793", "publisher": "University College London", "publication": "Mathematika", "publication_date": "2022-10", "series_number": "4", "volume": "68", "issue": "4", "pages": "1292-1301" }, { "id": "authors:jvfeg-gne85", "collection": "authors", "collection_id": "jvfeg-gne85", "cite_using_url": "https://authors.library.caltech.edu/records/jvfeg-gne85", "type": "conference_item", "title": "A Proof of the Kahn-Kalai Conjecture", "author": [ { "family_name": "Park", "given_name": "Jinyoung", "orcid": "0000-0003-3962-5668" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" } ], "abstract": "Proving the \"expectation-threshold\" conjecture of Kahn and Kalai, we show that for any increasing property $\\mathcal{F}$ on a finite set X, \\begin{equation*}p_{c}(\\mathcal{F})=O(q(\\mathcal{F})\\log\\ell(\\mathcal{F})),\\end{equation*} where $p_{c}(\\mathcal{F})$ and $q(\\mathcal{F})$ are the threshold and \"expectation threshold\" of $\\mathcal{F}$, and $\\ell(\\mathcal{F})$ is the maximum size of a minimal member of $\\mathcal{F}$.", "doi": "10.1109/focs54457.2022.00066", "publisher": "IEEE", "publication": "2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)", "publication_date": "2022-10", "pages": "636-639" }, { "id": "authors:b80br-qcv76", "collection": "authors", "collection_id": "b80br-qcv76", "cite_using_url": "https://authors.library.caltech.edu/records/b80br-qcv76", "type": "article", "title": "Tower-type bounds for Roth's theorem with popular differences", "author": [ { "family_name": "Fox", "given_name": "Jacob" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Zhao", "given_name": "Yufei", "orcid": "0000-0002-1995-3755" } ], "abstract": "Green developed an arithmetic regularity lemma to prove a strengthening of Roth's theorem on arithmetic progressions in dense sets. It states that for every\n \n \\epsilon > 0\n \n there is some\n \n N_0(\\epsilon)\n \n such that for every\n \n N \\ge N_0(\\epsilon)\n \n and\n \n A \\subset [N]\n \n with\n \n |A| = \\alpha N\n \n , there is some nonzero\n \n d\n \n such that\n \n A\n \n contains at least\n \n (\\alpha^3 - \\epsilon) N\n \n three-term arithmetic progressions with common difference\n \n d\n \n .\n \n \n We prove that the minimum\n \n N_0(\\epsilon)\n \n in Green's theorem is an exponential tower of twos of height on the order of\n \n \\log(1/\\epsilon)\n \n . Both the lower and upper bounds are new. This shows that the tower-type bounds that arise from the use of a regularity lemma in this application are quantitatively necessary.", "doi": "10.4171/jems/1271", "issn": "1435-9855", "publisher": "European Mathematical Society - EMS - Publishing House GmbH", "publication": "Journal of the European Mathematical Society", "publication_date": "2022-09-27", "series_number": "10", "volume": "25", "issue": "10", "pages": "3795-3831" }, { "id": "authors:wf08c-mwx40", "collection": "authors", "collection_id": "wf08c-mwx40", "cite_using_url": "https://authors.library.caltech.edu/records/wf08c-mwx40", "type": "article", "title": "Universality for Low-Degree Factors of Random Polynomials over Finite Fields", "author": [ { "family_name": "He", "given_name": "Jimmy", "orcid": "0000-0003-4345-6537" }, { "family_name": "Tuan Pham", "given_name": "Huy", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Wenqiang Xu", "given_name": "Max" } ], "abstract": "Abstract\n We show that the counts of low-degree irreducible factors of a random polynomial $f$ over $\\mathbb {F}_q$ with independent but nonuniform coefficients behave like that of a uniform random polynomial, exhibiting a form of universality for random polynomials over finite fields. Our strongest results require various assumptions on the parameters, but we are able to obtain results requiring only $q=p$ a prime with $p\\leq \\exp ({n^{1/13}})$ where $n$ is the degree of the polynomial. Our proofs use Fourier analysis and rely on tools recently applied by Breuillard and Varj\u00fa to study the $ax+b$ process, which show equidistribution for $f(\\alpha )$ at a single point. We extend this to handle multiple roots and the Hasse derivatives of $f$, which allow us to study the irreducible factors with multiplicity.", "doi": "10.1093/imrn/rnac239", "issn": "1073-7928", "publisher": "Oxford University Press (OUP)", "publication": "International Mathematics Research Notices", "publication_date": "2022-09-02", "series_number": "17", "volume": "2023", "issue": "17", "pages": "14752-14794" }, { "id": "authors:2wcx5-6zd15", "collection": "authors", "collection_id": "2wcx5-6zd15", "cite_using_url": "https://authors.library.caltech.edu/records/2wcx5-6zd15", "type": "article", "title": "Spectral independence, coupling, and the spectral gap of the Glauber dynamics", "author": [ { "family_name": "Jain", "given_name": "Vishesh", "orcid": "0000-0002-7275-3218" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Vuong", "given_name": "Thuy-Duong", "orcid": "0000-0003-0271-9687" } ], "abstract": "We present a new lower bound on the spectral gap of the Glauber dynamics for the Gibbs distribution of a spectrally independent q-spin system on a graph G = ( V , E ) with maximum degree \u0394. Our bound improves upon all previous bounds on the spectral gap obtained in this generality. Notably, for several interesting examples, we are able to handle the entire regime of \u0394 excluded by coupling arguments. As concrete applications, by combining our new lower bound with known spectral independence computations and known coupling arguments: (1) we show that for a triangle-free graph G = ( V , E ) with maximum degree \u0394 \u2265 3 , the Glauber dynamics for the uniform distribution on proper k-colorings with k \u2265 ( 1.763 \u2026 + \u03b4 ) \u0394 colors has spectral gap \u03a9 \u02dc \u03b4 ( | V | \u2212 1 ) . Previously, such a result was known either if the girth of G is at least 5 (Dyer et al. (2004) [12]), or under restrictions on \u0394 (Chen et al. (2021) [7]; Hayes and Vigoda (2003) [18]). (2) We show for the binomial random graph G ( n , d / n ) with d = O ( 1 ) , with high probability, an approximately uniformly random matching may be sampled in time O d ( n 2 + o ( 1 ) ) . This improves the corresponding running time of O \u02dc d ( n 3 ) due to Jerrum and Sinclair (1989) [22]; Jerrum (2003) [21].", "doi": "10.1016/j.ipl.2022.106268", "issn": "0020-0190", "publisher": "Elsevier", "publication": "Information Processing Letters", "publication_date": "2022-08", "volume": "177", "pages": "106268" }, { "id": "authors:kt7jj-7nn90", "collection": "authors", "collection_id": "kt7jj-7nn90", "cite_using_url": "https://authors.library.caltech.edu/records/kt7jj-7nn90", "type": "conference_item", "title": "Entropic independence: optimal mixing of down-up random walks", "author": [ { "family_name": "Anari", "given_name": "Nima", "orcid": "0000-0002-4394-3530" }, { "family_name": "Jain", "given_name": "Vishesh", "orcid": "0000-0002-7275-3218" }, { "family_name": "Koehler", "given_name": "Frederic", "orcid": "0000-0001-5220-9680" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Vuong", "given_name": "Thuy-Duong", "orcid": "0000-0003-0271-9687" } ], "abstract": "We introduce a notion called entropic independence that is an entropic analog of spectral notions of high-dimensional expansion. Informally, entropic independence of a background distribution \u00b5 on k-sized subsets of a ground set of elements says that for any (possibly randomly chosen) set S, the relative entropy of a single element of S drawn uniformly at random carries at most O(1/k) fraction of the relative entropy of S. Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mixing time bounds, overcoming the lossy nature of spectral analysis of Markov chains on exponential-sized state spaces. In our main technical result, we show a general way of deriving entropy contraction, a.k.a. modified log-Sobolev inequalities, for down-up random walks from spectral notions. We show that spectral independence of a distribution under arbitrary external fields automatically implies entropic independence. We furthermore extend our theory to the case where spectral independence does not hold under arbitrary external fields. To do this, we introduce a framework for obtaining tight mixing time bounds for Markov chains based on what we call restricted modified log-Sobolev inequalities, which guarantee entropy contraction not for all distributions, but for those in a sufficiently large neighborhood of the stationary distribution. To derive our results, we relate entropic independence to properties of polynomials: \u00b5 is entropically independent exactly when a transformed version of the generating polynomial of \u00b5 is upper bounded by its linear tangent; this property is implied by concavity of the said transformation, which was shown by prior work to be locally equivalent to spectral independence. We apply our results to obtain (1) tight modified log-Sobolev inequalities and mixing times for multi-step down-up walks on fractionally log-concave distributions, (2) the tight mixing time of O(nlogn) for Glauber dynamics on Ising models whose interaction matrix has eigenspectrum lying within an interval of length smaller than 1, improving upon the prior quadratic dependence on n, and (3) nearly-linear time O\u03b4(n) samplers for the hardcore and Ising models on n-node graphs that have \u03b4-relative gap to the tree-uniqueness threshold. In the last application, our bound on the running time does not depend on the maximum degree \u0394 of the graph, and is therefore optimal even for high-degree graphs, and in fact, is sublinear in the size of the graph for high-degree graphs.", "doi": "10.1145/3519935.3520048", "publisher": "ACM", "publication": "Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing", "publication_date": "2022-06-09", "pages": "1418-1430" }, { "id": "authors:g1cd6-4aq79", "collection": "authors", "collection_id": "g1cd6-4aq79", "cite_using_url": "https://authors.library.caltech.edu/records/g1cd6-4aq79", "type": "conference_item", "title": "Towards the sampling Lov\u00e1sz Local Lemma", "author": [ { "family_name": "Jain", "given_name": "Vishesh", "orcid": "0000-0002-7275-3218" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Vuong", "given_name": "Thuy Duong", "orcid": "0000-0003-0271-9687" } ], "abstract": "Let $\\Phi=(V, \\mathcal{C})$ be a constraint satisfaction problem on variables $v_{1}, \\ldots, v_{n}$ such that each constraint depends on at most $k$ variables and such that each variable assumes values in an alphabet of size at most [$q$]. Suppose that each constraint shares variables with at most $\\Delta$ constraints and that each constraint is violated with probability at most $p$ (under the product measure on its variables). We show that for $k, q=O(1)$, there is a deterministic, polynomial time algorithm to approximately count the number of satisfying assignments and a randomized, polynomial time algorithm to sample from approximately the uniform distribution on satisfying assignments, provided that $C\\cdot q^{2}\\cdot k\\cdot p\\cdot\\Delta^{7} < 1$, where $C$ is an absolute constant. Previously, a result of this form was known essentially only in the special case when each constraint is violated by exactly one assignment to its variables. For the special case of $k$ .CNF formulas, the term $\\Delta^{7}$ improves the previously best known $\\Delta^{60}$ for deterministic algorithms [Moitra, J.ACM, 2019] and $\\Delta^{13}$ for randomized algorithms [Feng, Guo, Yin, and Zhang, STOC 2021]. For the special case of properly $q$-coloring $k$-uniform hypergraphs, the term $\\Delta^{7}$ improves the previously best known $\\Delta^{14}$ for deterministic algorithms [Guo, Liao, Lu, and Zhang, SICOMP, 2019] and $\\Delta^{9}$ for randomized algorithms [Feng, Guo, Yin, and Zhang, STOC 2021].", "doi": "10.1109/focs52979.2021.00025", "publisher": "IEEE", "publication": "2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)", "publication_date": "2022-02", "pages": "173-183" }, { "id": "authors:rqnfa-1pg82", "collection": "authors", "collection_id": "rqnfa-1pg82", "cite_using_url": "https://authors.library.caltech.edu/records/rqnfa-1pg82", "type": "article", "title": "Mixing time of fractional random walk on finite fields", "author": [ { "family_name": "He", "given_name": "Jimmy", "orcid": "0000-0003-4345-6537" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Xu", "given_name": "Max Wenqiang" } ], "abstract": "We study a random walk on F_p defined by X_(n+1) = 1\u2215X_n + ε_(n+1) if X_n ≠ 0, and X_(n+1) = ε_(n+1) if X_n = 0, where ε_(n+1) are independent and identically distributed. This can be seen as a non-linear analogue of the Chung–Diaconis–Graham process. We show that the mixing time is of order log p, answering a question of Chatterjee and Diaconis.
", "doi": "10.1214/22-ejp858", "issn": "1083-6489", "publisher": "Institute of Mathematical Statistics", "publication": "Electronic Journal of Probability", "publication_date": "2022-01-01", "series_number": "none", "volume": "27", "issue": "none" }, { "id": "authors:jfxe1-xbm62", "collection": "authors", "collection_id": "jfxe1-xbm62", "cite_using_url": "https://authors.library.caltech.edu/records/jfxe1-xbm62", "type": "article", "title": "Common and Sidorenko Linear Equations", "author": [ { "family_name": "Fox", "given_name": "Jacob" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Zhao", "given_name": "Yufei", "orcid": "0000-0002-1995-3755" } ], "abstract": "A linear equation with coefficients in $\\mathbb{F}_q$ is common if the number of monochromatic solutions in any two-coloring of $\\mathbb{F}_q^{\\,n}$ is asymptotically (as $n \\to \\infty$) at least the number expected in a random two-coloring. The linear equation is Sidorenko if the number of solutions in any dense subset of $\\mathbb{F}_q^{\\,n}$ is asymptotically at least the number expected in a random set of the same density. In this paper, we characterize those linear equations which are common, and those which are Sidorenko. The main novelty is a construction based on choosing random Fourier coefficients that shows that certain linear equations do not have these properties. This solves problems posed in a paper of Saad and Wolf.
", "doi": "10.1093/qmath/haaa068", "issn": "0033-5606", "publisher": "Oxford University Press (OUP)", "publication": "The Quarterly Journal of Mathematics", "publication_date": "2021-01-20", "series_number": "4", "volume": "72", "issue": "4", "pages": "1223-1234" }, { "id": "authors:rznf7-sqp73", "collection": "authors", "collection_id": "rznf7-sqp73", "cite_using_url": "https://authors.library.caltech.edu/records/rznf7-sqp73", "type": "monograph", "title": "Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models", "author": [ { "family_name": "Anari", "given_name": "Nima", "orcid": "0000-0002-4394-3530" }, { "family_name": "Jain", "given_name": "Vishesh", "orcid": "0000-0002-7275-3218" }, { "family_name": "Koehler", "given_name": "Frederic", "orcid": "0000-0001-5220-9680" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Vuong", "given_name": "Thuy-Duong", "orcid": "0000-0003-0271-9687" } ], "abstract": "We introduce a notion called entropic independence that is an entropic analog of spectral notions of high-dimensional expansion. Informally, entropic independence of a background distribution $\u03bc$ on $k$-sized subsets of a ground set of elements says that for any (possibly randomly chosen) set $S$, the relative entropy of a single element of $S$ drawn uniformly at random carries at most $O(1/k)$ fraction of the relative entropy of $S$. Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mixing time bounds, overcoming the lossy nature of spectral analysis of Markov chains on exponential-sized state spaces. In our main technical result, we show a general way of deriving entropy contraction, a.k.a. modified log-Sobolev inequalities, for down-up random walks from spectral notions. We show that spectral independence of a distribution under arbitrary external fields automatically implies entropic independence. To derive our results, we relate entropic independence to properties of polynomials: $\u03bc$ is entropically independent exactly when a transformed version of the generating polynomial of $\u03bc$ is upper bounded by its linear tangent; this property is implied by concavity of the said transformation, which was shown by prior work to be locally equivalent to spectral independence. We apply our results to obtain tight modified log-Sobolev inequalities and mixing times for multi-step down-up walks on fractionally log-concave distributions. As our flagship application, we establish the tight mixing time of $O(n\\log n)$ for Glauber dynamics on Ising models whose interaction matrix has eigenspectrum lying within an interval of length smaller than $1$, improving upon the prior quadratic dependence on $n$.", "doi": "10.48550/arxiv.2106.04105", "publisher": "arXiv", "publication_date": "2021" }, { "id": "authors:aensr-td180", "collection": "authors", "collection_id": "aensr-td180", "cite_using_url": "https://authors.library.caltech.edu/records/aensr-td180", "type": "monograph", "title": "Entropic Independence II: Optimal Sampling and Concentration via Restricted Modified Log-Sobolev Inequalities", "author": [ { "family_name": "Anari", "given_name": "Nima", "orcid": "0000-0002-4394-3530" }, { "family_name": "Jain", "given_name": "Vishesh", "orcid": "0000-0002-7275-3218" }, { "family_name": "Koehler", "given_name": "Frederic", "orcid": "0000-0001-5220-9680" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" }, { "family_name": "Vuong", "given_name": "Thuy-Duong", "orcid": "0000-0003-0271-9687" } ], "abstract": "We introduce a framework for obtaining tight mixing times for Markov chains based on what we call restricted modified log-Sobolev inequalities. Modified log-Sobolev inequalities (MLSI) quantify the rate of relative entropy contraction for the Markov operator, and are notoriously difficult to establish. However, infinitesimally close to stationarity, entropy contraction becomes equivalent to variance contraction, a.k.a. a Poincare inequality, which is significantly easier to establish through, e.g., spectral analysis. Motivated by this observation, we study restricted modified log-Sobolev inequalities that guarantee entropy contraction not for all starting distributions, but for those in a large neighborhood of the stationary distribution. We show how to sample from the hardcore and Ising models on $n$-node graphs that have a constant $\u03b4$ relative gap to the tree-uniqueness threshold, in nearly-linear time $\\widetilde O_\u03b4(n)$. Notably, our bound does not depend on the maximum degree $\u0394$, and is therefore optimal even for high-degree graphs. This improves on prior mixing time bounds of $\\widetilde O_{\u03b4, \u0394}(n)$ and $\\widetilde O_\u03b4(n^2)$, established via (non-restricted) modified log-Sobolev and Poincare inequalities respectively. We further show that optimal concentration inequalities can still be achieved from the restricted form of modified log-Sobolev inequalities. To establish restricted entropy contraction, we extend the entropic independence framework of Anari, Jain, Koehler, Pham, and Vuong to the setting of distributions that are spectrally independent under a restricted set of external fields. We also develop an orthogonal trick that might be of independent interest: utilizing Bernoulli factories we show how to implement Glauber dynamics updates on high-degree graphs in $O(1)$ time, assuming standard adjacency array representation of the graph.", "doi": "10.48550/arxiv.2111.03247", "publisher": "arXiv", "publication_date": "2021" } ]