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Wojciech Samotij

Publications and source records attributed to Wojciech Samotij.

At least 37 records · Page 2Linked to original sources
arXiv2020

An efficient container lemma

We prove a new, efficient version of the hypergraph container theorems that is suited for hypergraphs with large uniformities. The main novelty is a refined approach to constructing containers that employs simple ideas from high-dimensional convex geometry. The existence of smaller families of containers for independent sets in such hypergraphs, which is guaranteed by the new theorem, allows us to improve upon the best currently known bounds for several problems in extremal graph theory, discrete geometry, and Ramsey theory.

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arXiv2019

On the counting problem in inverse Littlewood--Offord theory

Let $ε_1, \dotsc, ε_n$ be i.i.d. Rademacher random variables taking values $\pm 1$ with probability $1/2$ each. Given an integer vector $\boldsymbol{a} = (a_1, \dotsc, a_n)$, its concentration probability is the quantity $ρ(\boldsymbol{a}):=\sup_{x\in \mathbb{Z}}\Pr(ε_1 a_1+\dots+ε_n a_n = x)$. The Littlewood-Offord problem asks for bounds on $ρ(\boldsymbol{a})$ under various hypotheses on $\boldsymbol{a}$, whereas the inverse Littlewood-Offord problem, posed by Tao and Vu, asks for a characterization of all vectors $\boldsymbol{a}$ for which $ρ(\boldsymbol{a})$ is large. In this paper, we study the associated counting problem: How many integer vectors $\boldsymbol{a}$ belonging to a specified set have large $ρ(\boldsymbol{a})$? The motivation for our study is that in typical applications, the inverse Littlewood-Offord theorems are only used to obtain such counting estimates. Using a more direct approach, we obtain significantly better bounds for this problem than those obtained using the inverse Littlewood--Offord theorems of Tao and Vu and of Nguyen and Vu. Moreover, we develop a framework for deriving upper bounds on the probability of singularity of random discrete matrices that utilizes our counting result. To illustrate the methods, we present the first `exponential-type' (i.e., $\exp(-n^c)$ for some positive constant $c$) upper bounds on the singularity probability for the following two models: (i) adjacency matrices of dense signed random regular digraphs, for which the previous best known bound is $O(n^{-1/4})$ due to Cook; and (ii) dense row-regular $\{0,1\}$-matrices, for which the previous best known bound is $O_{C}(n^{-C})$ for any constant $C>0$ due to Nguyen.

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arXiv2019

On the probability of nonexistence in binomial subsets

Given a hypergraph $Γ=(Ω,\mathcal{X})$ and a sequence $\mathbf{p} = (p_ω)_{ω\in Ω}$ of values in $(0,1)$, let $Ω_{\mathbf{p}}$ be the random subset of $Ω$ obtained by keeping every vertex $ω$ independently with probability $p_ω$. We investigate the general question of deriving fine (asymptotic) estimates for the probability that $Ω_{\mathbf{p}}$ is an independent set in $Γ$, which is an omnipresent problem in probabilistic combinatorics. Our main result provides a sequence of upper and lower bounds on this probability, each of which can be evaluated explicitly in terms of the joint cumulants of small sets of edge indicator random variables. Under certain natural conditions, these upper and lower bounds coincide asymptotically, thus giving the precise asymptotics of the probability in question. We demonstrate the applicability of our results with two concrete examples: subgraph containment in random (hyper)graphs and arithmetic progressions in random subsets of the integers.

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arXiv2019

A generalized Turán problem in random graphs

We study the following generalization of the Turán problem in sparse random graphs. Given graphs $T$ and $H$, let $\mathrm{ex}\big(G(n,p), T, H\big)$ be the random variable that counts the largest number of copies of $T$ in a subgraph of $G(n,p)$ that does not contain $H$. We study the threshold phenomena arising in the evolution of the typical value of this random variable, for every $H$ and an arbitrary $2$-balanced $T$. Our results in the case when $m_2(H) > m_2(T)$ are a natural generalization of the Erdős--Stone theorem for $G(n,p)$, which was proved several years ago by Conlon and Gowers and by Schacht; the case $T = K_m$ has been recently resolved by Alon, Kostochka, and Shikhelman. More interestingly, the case when $m_2(H) \le m_2(T)$ exhibits a more complex and subtle behavior. Namely, the location(s) of the (possibly multiple) threshold(s) are determined by densities of various coverings of $H$ with copies of $T$ and the typical value(s) of $\mathrm{ex}\big(G(n,p), T, H\big)$ are given by solutions to deterministic hypergraph Turán-type problems that we are unable to solve in full generality.

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arXiv2018

An asymmetric container lemma and the structure of graphs with no induced $4$-cycle

The method of hypergraph containers, introduced recently by Balogh, Morris, and Samotij, and independently by Saxton and Thomason, has proved to be an extremely useful tool in the study of various monotone graph properties. In particular, a fairly straightforward application of this technique allows one to locate, for each non-bipartite graph $H$, the threshold at which the distribution of edges in a typical $H$-free graph with a given number of edges undergoes a transition from 'random-like' to 'structured'. On the other hand, for non-monotone hereditary graph properties the standard version of this method does not allow one to establish even the existence of such a threshold. In this paper we introduce a refinement of the container method that takes into account the asymmetry between edges and non-edges in a sparse member of a hereditary graph property. As an application, we determine the approximate structure of a typical graph with $n$ vertices, $m$ edges, and no induced copy of the $4$-cycle, for each function $m = m(n)$ satisfying $n^{4/3} (\log n)^4 \leqslant m \ll n^2$. We show that almost all such graphs $G$ have the following property: the vertex set of $G$ can be partitioned into an 'almost-independent' set (a set with $o(m)$ edges) and an 'almost-clique' (a set inducing a subgraph with density $1-o(1)$). The lower bound on $m$ is optimal up to a polylogarithmic factor, as standard arguments show that if $n \ll m \ll n^{4/3}$, then almost all such graphs are 'random-like'. As a further consequence, we deduce that the random graph $G(n,p)$ conditioned to contain no induced $4$-cycles undergoes phase transitions at $p = n^{-2/3 + o(1)}$ and $p = n^{-1/3 + o(1)}$.

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arXiv2018

The method of hypergraph containers

In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite objects with forbidden substructures. This technique exploits a subtle clustering phenomenon exhibited by the independent sets of uniform hypergraphs whose edges are sufficiently evenly distributed; more precisely, it provides a relatively small family of 'containers' for the independent sets, each of which contains few edges. We attempt to convey to the reader a general high-level overview of the method, focusing on a small number of illustrative applications in areas such as extremal graph theory, Ramsey theory, additive combinatorics, and discrete geometry, and avoiding technical details as much as possible.

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arXiv2017

Supersaturated sparse graphs and hypergraphs

A central problem in extremal graph theory is to estimate, for a given graph $H$, the number of $H$-free graphs on a given set of $n$ vertices. In the case when $H$ is not bipartite, fairly precise estimates on this number are known. In particular, thirty years ago, Erdős, Frankl, and Rödl proved that there are $2^{(1+o(1))\text{ex}(n,H)}$ such graphs. In the bipartite case, however, nontrivial bounds have been proven only for relatively few special graphs $H$. We make a first attempt at addressing this enumeration problem for a general bipartite graph $H$. We show that an upper bound of $2^{O(\text{ex}(n,H))}$ on the number of $H$-free graphs with $n$ vertices follows merely from a rather natural assumption on the growth rate of $n \mapsto \text{ex}(n,H)$; an analogous statement remains true when $H$ is a uniform hypergraph. Subsequently, we derive several new results, along with most previously known estimates, as simple corollaries of our theorem. At the heart of our proof lies a general supersaturation statement that extends the seminal work of Erdős and Simonovits. The bounds on the number of $H$-free hypergraphs are derived from it using the method of hypergraph containers.

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arXiv2017

Subsets of posets minimising the number of chains

A well-known theorem of Sperner describes the largest collections of subsets of an $n$-element set none of which contains another set from the collection. Generalising this result, Erdős characterised the largest families of subsets of an $n$-element set that do not contain a chain of sets $A_1 \subset \dotsc \subset A_k$ of an arbitrary length $k$. The extremal families contain all subsets whose cardinalities belong to an interval of length $k-1$ centred at $n/2$. In a far-reaching extension of Sperner's theorem, Kleitman determined the smallest number of chains of length two that have to appear in a collection of a given number $a$ of subsets of an $n$-element set. For every $a$, this minimum is achieved by the collection comprising $a$ sets whose cardinalities are as close to $n/2+1/4$ as possible. We show that the same is true about chains of an arbitrary length $k$, for all $a$ and $n$, confirming the prediction Kleitman made fifty years ago. We also characterise all families of $a$ subsets with the smallest number of chains of length $k$ for all $a$ for which this smallest number is positive. Our argument is inspired by an elegant probabilistic lemma from a recent paper of Noel, Scott, and Sudakov, which in turn can be traced back to Lubell's proof of Sperner's theorem.

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arXiv2016

Exponential decay of loop lengths in the loop $O(n)$ model with large $n$

The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been conjectured that both the spin and the loop $O(n)$ models exhibit exponential decay of correlations when $n>2$. We verify this for the loop $O(n)$ model with large parameter $n$, showing that long loops are exponentially unlikely to occur, uniformly in the edge weight $x$. Our proof provides further detail on the structure of typical configurations in this regime. Putting appropriate boundary conditions, when $nx^6$ is sufficiently small, the model is in a dilute, disordered phase in which each vertex is unlikely to be surrounded by any loops, whereas when $nx^6$ is sufficiently large, the model is in a dense, ordered phase which is a small perturbation of one of the three ground states.

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arXiv2016

Packing trees of unbounded degrees in random graphs

In this paper, we address the problem of packing large trees in $G_{n,p}$. In particular, we prove the following result. Suppose that $T_1, \dotsc, T_N$ are $n$-vertex trees, each of which has maximum degree at most $(np)^{1/6} / (\log n)^6$. Then with high probability, one can find edge-disjoint copies of all the $T_i$ in the random graph $G_{n,p}$, provided that $p \geq (\log n)^{36}/n$ and $N \le (1-\varepsilon)np/2$ for a positive constant $\varepsilon$. Moreover, if each $T_i$ has at most $(1-α)n$ vertices, for some positive $α$, then the same result holds under the much weaker assumptions that $p \geq (\log n)^2/(cn)$ and $Δ(T_i) \leq c np / \log n$ for some~$c$ that depends only on $α$ and $\varepsilon$. Our assumptions on maximum degrees of the trees are significantly weaker than those in all previously known approximate packing results.

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arXiv2016

The number of additive triples in subsets of abelian groups

A set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elements $x,y,z$ with $x+y=z$. The study of how large the largest sum-free subset of a given abelian group is had started more than thirty years before it was finally resolved by Green and Ruzsa a decade ago. We address the following more general question. Suppose that a set $A$ of elements of an abelian group $G$ has cardinality $a$. How many Schur triples must $A$ contain? Moreover, which sets of $a$ elements of $G$ have the smallest number of Schur triples? In this paper, we answer these questions for various groups $G$ and ranges of $a$.

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arXiv2015

Smoothed analysis on connected graphs

The main paradigm of smoothed analysis on graphs suggests that for any large graph $G$ in a certain class of graphs, perturbing slightly the edges of $G$ at random (usually adding few random edges to $G$) typically results in a graph having much "nicer" properties. In this work we study smoothed analysis on trees or, equivalently, on connected graphs. Given an $n$-vertex connected graph $G$, form a random supergraph $G^*$ of $G$ by turning every pair of vertices of $G$ into an edge with probability $\fracε{n}$, where $ε$ is a small positive constant. This perturbation model has been studied previously in several contexts, including smoothed analysis, small world networks, and combinatorics. Connected graphs can be bad expanders, can have very large diameter, and possibly contain no long paths. In contrast, we show that if $G$ is an $n$-vertex connected graph then typically $G^*$ has edge expansion $Ω(\frac{1}{\log n})$, diameter $O(\log n)$, vertex expansion $Ω(\frac{1}{\log n})$, and contains a path of length $Ω(n)$, where for the last two properties we additionally assume that $G$ has bounded maximum degree. Moreover, we show that if $G$ has bounded degeneracy, then typically the mixing time of the lazy random walk on $G^*$ is $O(\log^2 n)$. All these results are asymptotically tight.

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arXiv2014

Counting independent sets in graphs

In this short survey article, we present an elementary, yet quite powerful, method of enumerating independent sets in graphs. This method was first employed more than three decades ago by Kleitman and Winston and has subsequently been used numerous times by many researchers in various contexts. Our presentation of the method is illustrated with several applications of it to `real-life' combinatorial problems. In particular, we derive bounds on the number of independent sets in regular graphs, sum-free subsets of $\{1, \ldots, n\}$, and $C_4$-free graphs and give a short proof of an analogue of Roth's theorem on $3$-term arithmetic progressions in sparse random sets of integers which was originally formulated and proved by Kohayakawa, Łuczak, and Rödl.

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arXiv2014

The typical structure of sparse $K_{r+1}$-free graphs

Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erdős and Rényi, and the family of $H$-free graphs, that is, graphs which do not contain a subgraph isomorphic to a given (usually small) graph $H$. A widely studied problem that lies at the interface of these two areas is that of determining how the structure of a typical $H$-free graph with $n$ vertices and $m$ edges changes as $m$ grows from $0$ to $\text{ex}(n,H)$. In this paper, we resolve this problem in the case when $H$ is a clique, extending a classical result of Kolaitis, Prömel, and Rothschild. In particular, we prove that for every $r \ge 2$, there is an explicit constant $θ_r$ such that, letting $m_r = θ_r n^{2-\frac{2}{r+2}} (\log n)^{1/\left[\binom{r+1}{2}-1\right]}$, the following holds for every positive constant $\varepsilon$. If $m \ge (1+\varepsilon) m_r$, then almost all $K_{r+1}$-free $n$-vertex graphs with $m$ edges are $r$-partite, whereas if $n \ll m \le (1-\varepsilon)m_r$, then almost all of them are not $r$-partite.

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arXiv2014

On the number of monotone sequences

One of the most classical results in Ramsey theory is the theorem of Erdős and Szekeres from 1935, which says that every sequence of more than $k^2$ numbers contains a monotone subsequence of length $k+1$. We address the following natural question motivated by this result: Given integers $k$ and $n$ with $n \geq k^2+1$, how many monotone subsequences of length $k+1$ must every sequence of $n$ numbers contain? We answer this question precisely for all sufficiently large $k$ and $n \leq k^2 + c k^{3/2} / \log k$, where $c$ is some absolute positive constant.

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arXiv2014

Independent sets in hypergraphs

Many important theorems in combinatorics, such as Szemerédi's theorem on arithmetic progressions and the Erdős-Stone Theorem in extremal graph theory, can be phrased as statements about independent sets in uniform hypergraphs. In recent years, an important trend in the area has been to extend such classical results to the so-called sparse random setting. This line of research culminated recently in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving problems of this type. In this paper, we provide a third, completely different approach to proving extremal and structural results in sparse random sets. We give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets. We then derive many interesting results as fairly straightforward consequences of this abstract theorem. In particular, we prove the well-known conjecture of Kohayakawa, Łuczak and Rödl, a probabilistic embedding lemma for sparse graphs. We also give alternative proofs of many of the results of Conlon and Gowers and Schacht, and obtain their natural counting versions, which in some cases are considerably stronger. We moreover prove a sparse version of the Erdős-Frankl-Rödl Theorem on the number of H-free graphs and extend a result of Rödl and Ruciński on Ramsey properties in sparse random graphs to the general, non-symmetric setting. We remark that similar results have been discovered independently by Saxton and Thomason, and that, in parallel to this work, Conlon, Gowers, Samotij and Schacht have proved a sparse analogue of the counting lemma for subgraphs of the random graph G(n,p), which may be viewed as a version of the KŁR conjecture that is stronger in some ways and weaker in others.

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arXiv2014

Long paths and cycles in random subgraphs of H-free graphs

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random subgraph $G_p$ of $G$ by independently keeping each edge of $G$ with probability $p$. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive $\varepsilon$, there exists a positive $δ$ (depending only on $\varepsilon$) such that the following holds: If $p \ge \frac{1+\varepsilon}{k}$, then with probability tending to $1$ as $k \to \infty$, the random graph $G_p$ contains a cycle of length at least $n_{\mathcal{H}}(δk)$, where $n_{\mathcal{H}}(k)>k$ is the minimum number of vertices in an $\mathcal{H}$-free graph of average degree at least $k$. Thus in particular $G_p$ as above typically contains a cycle of length at least linear in $k$.

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