[ { "id": "authors:2vghy-wsp47", "collection": "authors", "collection_id": "2vghy-wsp47", "cite_using_url": "https://authors.library.caltech.edu/records/2vghy-wsp47", "type": "monograph", "title": "Sunflowers in set systems with small VC-dimension", "author": [ { "family_name": "Balogh", "given_name": "J\u00f3zsef" }, { "family_name": "Bernshteyn", "given_name": "Anton" }, { "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 $\\{A_1,\\ldots, A_r\\}$ is an $r$-sunflower if for all $1 \\leqslant i < j \\leqslant r$ and $1 \\leqslant i' < j' \\leqslant r$, we have $A_i \\cap A_j = A_{i'} \\cap A_{j'}$. Erd\u0151s and Rado conjectured in 1960 that every family $\\mathcal{H}$ of $\\ell$-element sets of size at least $K(r)^\\ell$ contains an $r$-sunflower, where $K(r)$ is some function that depends only on $r$. We prove that if $\\mathcal{H}$ is a family of $\\ell$-element sets of VC-dimension at most $d$ and $|\\mathcal{H}| > (C r (\\log d+\\log^\\ast \\ell))^\\ell$ for some absolute constant $C > 0$, then $\\mathcal{H}$ contains an $r$-sunflower. This improves a recent result of Fox, Pach, and Suk. When $d=1$, we obtain a sharp bound, namely that $|\\mathcal{H}| > (r-1)^\\ell$ 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": "10.48550/arxiv.2408.04165", "publisher": "arXiv", "publication_date": "2024" }, { "id": "authors:3c516-6z150", "collection": "authors", "collection_id": "3c516-6z150", "cite_using_url": "https://authors.library.caltech.edu/records/3c516-6z150", "type": "monograph", "title": "Sharp bound for the Erd\u0151s-Straus 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 $\\{1, \\ldots, n\\}$ 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": "10.48550/arxiv.2410.14624", "publisher": "arXiv", "publication_date": "2024" }, { "id": "authors:jzzp0-cpa20", "collection": "authors", "collection_id": "jzzp0-cpa20", "cite_using_url": "https://authors.library.caltech.edu/records/jzzp0-cpa20", "type": "monograph", "title": "Sampling from Spherical Spin Glasses in Total Variation via Algorithmic Stochastic Localization", "author": [ { "family_name": "Huang", "given_name": "Brice" }, { "family_name": "Montanari", "given_name": "Andrea" }, { "family_name": "Pham", "given_name": "Huy Tuan", "orcid": "0000-0003-4659-4345", "clpid": "Pham-Huy-Tuan" } ], "abstract": "We consider the problem of algorithmically sampling from the Gibbs measure of a mixed $p$-spin spherical spin glass. We give a polynomial-time algorithm that samples from the Gibbs measure up to vanishing total variation error, for any model whose mixture satisfies $$\u03be''(s) < \\frac{1}{(1-s)^2}, \\qquad \\forall s\\in [0,1).$$ This includes the pure $p$-spin glasses above a critical temperature that is within an absolute ($p$-independent) constant of the so-called shattering phase transition. Our algorithm follows the algorithmic stochastic localization approach introduced in (Alaoui, Montanari, Sellke, 20022). A key step of this approach is to estimate the mean of a sequence of tilted measures. We produce an improved estimator for this task by identifying a suitable correction to the TAP fixed point selected by approximate message passing (AMP). As a consequence, we improve the algorithm's guarantee over previous work, from normalized Wasserstein to total variation error. In particular, the new algorithm and analysis opens the way to perform inference about one-dimensional projections of the measure.", "doi": "10.48550/arxiv.2404.15651", "publisher": "arXiv", "publication_date": "2024" }, { "id": "authors:nh023-sdw60", "collection": "authors", "collection_id": "nh023-sdw60", "cite_using_url": "https://authors.library.caltech.edu/records/nh023-sdw60", "type": "monograph", "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 Tur\u00e1n-type question. Suppose $\u0393$ 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 $G \\subseteq \u0393$ which does not contain a fixed graph $H$ as an induced subgraph or, more generally, which belongs to a hereditary property $\\mathcal{P}$? This provides a common generalization of two recently studied cases, namely $\u0393$ 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.\n We determine the answer up to a constant factor with respect to $n$ and $t$, for certain bipartite $H$ and for $\u0393$ 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 $\u0393$. 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": "10.48550/arxiv.2405.05902", "publisher": "arXiv", "publication_date": "2024" }, { "id": "authors:w3v6a-pqr34", "collection": "authors", "collection_id": "w3v6a-pqr34", "cite_using_url": "https://authors.library.caltech.edu/records/w3v6a-pqr34", "type": "monograph", "title": "On the clique number of random Cayley graphs and related topics", "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" }, { "family_name": "Yepremyan", "given_name": "Liana" } ], "abstract": "We prove that a random Cayley graph on a group of order $N$ has clique number $O(\\log N \\log \\log N)$ with high probability. This bound is best possible up to the constant factor for certain groups, including~$\\mathbb{F}_2^n$, and improves the longstanding upper bound of $O(\\log^2 N)$ due to Alon. Our proof does not make use of the underlying group structure and is purely combinatorial, with the key result being an essentially best possible upper bound for the number of subsets of given order that contain at most a given number of colors in a properly edge-colored complete graph. As a further application of this result, we study a conjecture of Alon stating that every group of order $N$ has a Cayley graph whose clique number and independence number are both $O(\\log N)$, proving the conjecture for all abelian groups of order $N$ for almost all $N$. For finite vector spaces of order $N$ with characteristic congruent to $1 \\pmod 4$, we prove the existence of a self-complementary Cayley graph on the vector space whose clique number and independence number are both at most $(2+o(1))\\log N$. This matches the lower bound for Ramsey numbers coming from random graphs and solves, in a strong form, a problem of Alon and Orlitsky motivated by information theory.", "doi": "10.48550/arxiv.2412.21194", "publisher": "arXiv", "publication_date": "2024" }, { "id": "authors:aydgt-f5883", "collection": "authors", "collection_id": "aydgt-f5883", "cite_using_url": "https://authors.library.caltech.edu/records/aydgt-f5883", "type": "monograph", "title": "Spread blow-up lemma with an application to perturbed random graphs", "author": [ { "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": "Combining ideas of Pham, Sah, Sawhney, and Simkin on spread perfect matchings in super-regular bipartite graphs with an algorithmic blow-up lemma, we prove a spread version of the blow-up lemma. Intuitively, this means that there exists a probability measure over copies of a desired spanning graph $H$ in a given system of super-regular pairs which does not heavily pin down any subset of vertices. This allows one to complement the use of the blow-up lemma with the recently resolved Kahn-Kalai conjecture. As an application, we prove an approximate version of a conjecture of B\u00f6ttcher, Parczyk, Sgueglia, and Skokan on the threshold for appearance of powers of Hamilton cycles in perturbed random graphs.", "doi": "10.48550/arxiv.2410.06132", "publisher": "arXiv", "publication_date": "2024" }, { "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: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" }, { "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" } ]