arXiv subjects
Publications and source records attributed to Pandelis Dodos.
It is an important fact that extremal discrete structures -- that is, discrete structures of maximal size among those that avoid certain configurations -- exhibit strong pseudorandom behavior. We present instances of this phenomenon in the context of \emph{graph-codes}, a notion put forth recently by Alon, as well as on problems related to the density polynomial Hales--Jewett conjecture.
This article obtains purely metric counterparts of cornerstone results in the theory of embedding graphs into normed spaces. Our first main result is a metric analogue of Matou\v{s}ek's extrapolation relating the Poincar\'e constants $\gamma(G,\varrho^p)$ and $\gamma(G,\varrho^q)$ for any exponents $0 < p,q < \infty$, any bounded-degree expander graph $G$, and any target metric space $\mathcal{M}=(M,\varrho)$. Our second main result provides a sharp estimate of the Poincar\'e constant $\gamma(G,\varrho)$ in terms of the cardinalities of the vertex set of $G$ and the metric space $\mathcal{M}=(M,\varrho)$, in the setting of \textit{random} graphs. This yields optimal estimates on the minimum cardinality of (bi-Lipschitz) universal metric spaces for graphs, finally establishing a nonlinear analogue of Matou\v{s}ek's celebrated "incompressibility" theorem (1996). Further, we obtain estimates on the nonlinear spectral gap of metric snowflakes and sharp lower bounds on the distortion of random regular graphs into arbitrary metric spaces. Our proofs develop new nonlinear techniques, including random compression methods and a novel structural dichotomy for metric embeddings.
Nonlinear Poincar\'e inequalities are indispensable tools in the study of dimension reduction and low-distortion embeddings of graphs into metric spaces, and have found remarkable algorithmic applications. A basic open problem, posed by Jon Kleinberg (2013), asks whether the optimal nonlinear Poincar\'e constant for maps between two independent $3$-regular random graphs is dimension-free, i.e., independent of vertex-set sizes. We give a complete and affirmative resolution to Kleinberg's problem, also allowing for arbitrary graph degrees. As a corollary, we obtain a stochastic construction of $O(1)\text{-universal}$ approximators for random graphs, answering a question of Mendel and Naor.
A graph $G=(V,E)$ is geometrically embeddable into a normed space $X$ when there is a mapping $\zeta: V\to X$ such that $\|\zeta(v)-\zeta(w)\|_X\leqslant 1$ if and only if $\{v,w\}\in E$, for all distinct $v,w\in V$. Our result is the following universal threshold for the embeddability of trees. Let $\Delta \geqslant 3$, and let $N$ be sufficiently large in terms of $\Delta$. Every $N$--vertex tree of maximal degree at most $\Delta$ is embeddable into any normed space of dimension at least $64\,\frac{\log N}{\log\log N}$, and complete trees are non-embeddable into any normed space of dimension less than $\frac{1}{2}\,\frac{\log N}{\log\log N}$. In striking contrast, spectral expanders and random graphs are known to be non-embeddable in sublogarithmic dimension. Our result is based on a randomized embedding whose analysis utilizes the recent breakthroughs on Bourgain's slicing problem.
A seminal open question of Pisier and Mendel--Naor asks whether every degree-regular graph which satisfies the classical discrete Poincar\'e inequality for scalar functions, also satisfies an analogous inequality for functions taking values in \textit{any} normed space with non-trivial cotype. Motivated by applications, it is also greatly important to quantify the dependence of the corresponding optimal Poincar\'e constant on the cotype $q$. Works of Odell--Schlumprecht (1994), Ozawa (2004), and Naor (2014) make substantial progress on the former question by providing a positive answer for normed spaces which also have an unconditional basis, in addition to finite cotype. However, little is known in the way of quantitative estimates: the mentioned results imply a bound on the Poincar\'e constant depending super-exponentially on $q$. We introduce a novel combinatorial framework for proving quantitative nonlinear spectral gap estimates. The centerpiece is a property of regular graphs that we call \emph{long range expansion}, which holds with high probability for random regular graphs. Our main result is that any regular graph with the long-range expansion property satisfies a discrete Poincar\'{e} inequality for any normed space with an unconditional basis and cotype $q$, with a Poincar\'{e} constant that depends \emph{polynomially} on $q$, which is optimal. As an application, any normed space with an unconditional basis which admits a low distortion embedding of an $n$-vertex random regular graph, must have cotype at least polylogarithmic in $n$. This extends a celebrated lower-bound of Matou\v{s}ek for low distortion embeddings of random graphs into $\ell_q$ spaces.
Let $n$ be a positive integer, let $0<p\leqslant p'\leqslant \frac{1}{2}$, and let $\ell \leqslant pn$ be a nonnegative integer. We prove that if $\mathcal{F},\mathcal{G}\subseteq \{0,1\}^n$ are two families whose cross intersections forbid $\ell$ -- that is, they satisfy $|A\cap B|\neq \ell$ for every $A\in\mathcal{F}$ and every $B\in\mathcal{G}$ -- then, setting $t:=\min\{\ell,pn-\ell\}$, we have the subgaussian bound \[ \mu_p(\mathcal{F})\, \mu_{p'}(\mathcal{G})\leqslant 2\exp\Big( - \frac{t^2}{58^2\,pn}\Big), \] where $\mu_p$ and $\mu_{p'}$ denote the $p$-biased and $p'$-biased measures on $\{0,1\}^n$ respectively.
We obtain estimates for the Kolmogorov distance to appropriately chosen gaussians, of linear functions \[ \sum_{i\in [n]^d} \theta_i X_i \] of random tensors $\boldsymbol{X}=\langle X_i:i\in [n]^d\rangle$ which are symmetric and exchangeable, and whose entries have bounded third moment and vanish on diagonal indices. These estimates are expressed in terms of intrinsic (and easily computable) parameters associated with the random tensor $\boldsymbol{X}$ and the given coefficients $\langle \theta_i:i\in [n]^d\rangle$, and they are optimal in various regimes. The key ingredient -- which is of independent interest -- is a combinatorial CLT for high-dimensional tensors which provides quantitative non-asymptotic normality under suitable conditions, of statistics of the form \[ \sum_{(i_1,\dots,i_d)\in [n]^d} \boldsymbol{\zeta}\big(i_1,\dots,i_d,\pi(i_1),\dots,\pi(i_d)\big) \] where $\boldsymbol{\zeta}\colon [n]^d\times [n]^d\to\mathbb{R}$ is a deterministic real tensor, and $\pi$ is a random permutation uniformly distributed on the symmetric group $\mathbb{S}_n$. Our results extend, in any dimension $d$, classical work of Bolthausen who covered the one-dimensional case, and more recent work of Barbour/Chen who treated the two-dimensional case.
A $d$-dimensional random array on a nonempty set $I$ is a stochastic process $\boldsymbol{X}=\langle X_s:s\in \binom{I}{d}\rangle$ indexed by the set $\binom{I}{d}$ of all $d$-element subsets of $I$. We obtain structural decompositions of finite, high-dimensional random arrays whose distribution is invariant under certain symmetries. Our first main result is a distributional decomposition of finite, (approximately) spreadable, high-dimensional random arrays whose entries take values in a finite set; the two-dimensional case of this result is the finite version of an infinitary decomposition due to Fremlin and Talagrand. Our second main result is a physical decomposition of finite, spreadable, high-dimensional random arrays with square-integrable entries that is the analogue of the Hoeffding/Efron--Stein decomposition. All proofs are effective. We also present applications of these decompositions in the study of concentration of functions of finite, high-dimensional random arrays.
Let $\boldsymbol{X}$ be a $d$-dimensional random array on $[n]$ whose entries take values in a finite set $\mathcal{X}$, that is, $\boldsymbol{X}=\langle X_s:s\in \binom{[n]}{d}\rangle$ is an $\mathcal{X}$-valued stochastic process indexed by the set $\binom{[n]}{d}$ of all $d$-element subsets of $[n]:=\{1,\dots,n\}$. We give easily checked conditions on $\boldsymbol{X}$ that ensure, for instance, that for every function $f\colon \mathcal{X}^{\binom{[n]}{d}}\to\mathbb{R}$ that satisfies $\mathbb{E}[f(\boldsymbol{X})]=0$ and $\|f(\boldsymbol{X})\|_{L_p}=1$ for some $p>1$, the random variable $f(\boldsymbol{X})$ becomes concentrated after conditioning it on a large subarray of $\boldsymbol{X}$. These conditions cover several classes of random arrays with not necessarily independent entries. Applications are given in combinatorics, and examples are also presented that show the optimality of various aspects of the results.
Let $n$ be a positive integer, let $\boldsymbol{X}=(X_1,\dots,X_n)$ be a random vector in $\mathbb{R}^n$ with bounded entries, and let $(\theta_1,\dots,\theta_n)$ be a vector in $\mathbb{R}^n$. We show that the subgaussian behavior of the random variable $\theta_1 X_1+\dots +\theta_n X_n$ is essentially determined by the subgaussian behavior of the random variables $\sum_{i\in H} \theta_i X_i$ where $H$ is a random subset of $\{1,\dots,n\}$.
Let $A$ be a finite set with $|A|\geqslant 2$, let $n$ be a positive integer, and let $A^n$ denote the discrete $n$-dimensional hypercube (that is, $A^n$ is the Cartesian product of $n$ many copies of $A$). Given a family $\langle D_t:t\in A^n\rangle$ of measurable events in a probability space (a stochastic process), what structural information can be obtained assuming that the events $\langle D_t:t\in A^n\rangle$ are not behaving as if they were independent? We obtain an answer to this problem (in a strong quantitative sense) subject to a mild "stationarity" condition. Our result has a number of combinatorial consequences, including a new (and the most informative so far) proof of the density Hales--Jewett theorem.
We show that, under some mild hypotheses, the Gowers uniformity norms (both in the additive and in the hypergraph setting) are essentially equivalent to certain weaker norms which are easier to understand. We present two applications of this equivalence: a variant of the Koopman--von Neumann decomposition, and a proof of the relative inverse theorem for the Gowers $U^s[N]$-norm using a norm-type pseudorandomness condition.
We prove a concentration inequality which asserts that, under some mild regularity conditions, every random variable defined on the product of sufficiently many probability spaces exhibits pseudorandom behavior.
We prove a variant of the abstract probabilistic version of Szemerédi's regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random variables in $L_p$ for any $p>1$. Our approach is based on martingale difference sequences.
We study sparse hypergraphs which satisfy a mild pseudorandomness condition known as $L_p$ regularity. We prove appropriate regularity and counting lemmas, and we extend the relative removal lemma of Tao in this setting. This answers a question of Borgs, Chayes, Cohn and Zhao.
We consider some variants of the Gowers box norms, introduced by Hatami, and show their relevance in the context of sparse hypergraphs. Our main results are the following. Firstly, we prove a generalized von Neumann theorem for $L_p$ graphons. Secondly, we give natural examples of pseudorandom families, that is, sparse weighted uniform hypergraphs which satisfy relative versions of the counting and removal lemmas.
We prove a density version of the Carlson--Simpson Theorem. Specifically we show the following. For every integer $k\geq 2$ and every set $A$ of words over $k$ satisfying \[\limsup_{n\to\infty} \frac{|A\cap [k]^n|}{k^n}>0\] there exist a word $c$ over $k$ and a sequence $(w_n)$ of left variable words over $k$ such that the set \[\{c\}\cup \big\{c^{\smallfrown}w_0(a_0)^{\smallfrown}...^{\smallfrown}w_n(a_n) : n\in\mathbb{N} \ \text{ and } \ a_0,...,a_n\in [k]\big\}\] is contained in $A$. While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.
For every integer $k\geq 2$ let $[k]^{<\mathbb{N}}$ be the set of all words over $k$, that is, all finite sequences having values in $[k]:=\{1,...,k\}$. A Carlson-Simpson tree of $[k]^{<\mathbb{N}}$ of dimension $m\geq 1$ is a subset of $[k]^{<\mathbb{N}}$ of the form \[ \{w\}\cup \big\{w^{\smallfrown}w_0(a_0)^{\smallfrown}...^{\smallfrown}w_{n}(a_n): n\in \{0,...,m-1\} \text{ and } a_0,...,a_n\in [k]\big\} \] where $w$ is a word over $k$ and $(w_n)_{n=0}^{m-1}$ is a finite sequence of left variable words over $k$. We study the behavior of a family of measurable events in a probability space indexed by the elements of a Carlson-Simpson tree of sufficiently large dimension. Specifically we show the following. For every integer $k\geq 2$, every $0<\varepsilon\leq 1$ and every integer $n\geq 1$ there exists a strictly positive constant $θ(k,\varepsilon,n)$ with the following property. If $m$ is a given positive integer, then there exists an integer $\mathrm{Cor}(k,\varepsilon,m)$ such that for every Carlson--Simpson tree $T$ of $[k]^{<\mathbb{N}}$ of dimension at least $\mathrm{Cor}(k,\varepsilon,m)$ and every family $\{A_t:t\in T\}$ of measurable events in a probability space $(Ω,Σ,μ)$ satisfying $μ(A_t)\geq \varepsilon$ for every $t\in T$, there exists a Carlson--Simpson tree $S$ of dimension $m$ with $S\subseteq T$ and such that for every nonempty $F\subseteq S$ we have \[μ\Big(\bigcap_{t\in F} A_t\Big) \geq θ(k,\varepsilon,|F|). \] The proof is based, among others, on the density version of the Carlson--Simpson Theorem established recently by the authors, as well as, on a partition result -- of independent interest -- closely related to the work of T. J. Carlson, and H. Furstenberg and Y. Katznelson. The argument is effective and yields explicit lower bounds for the constants $θ(k,\varepsilon,n)$.