arXiv subjects

Matthew Kwan

Publications and source records attributed to Matthew Kwan.

At least 55 records · Page 3Linked to original sources

arXiv2021

Ramsey graphs induce subgraphs of quadratically many sizes

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in fact all Ramsey graphs must obey certain "richness" properties characteristic of random graphs. Motivated by an old problem of Erdős and McKay, recently Narayanan, Sahasrabudhe and Tomon conjectured that for any fixed C, every n-vertex C-Ramsey graph induces subgraphs of $Θ(n^2)$ different sizes. In this paper we prove this conjecture.

math.CO↗

arXiv2021

Proof of a conjecture on induced subgraphs of Ramsey graphs

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in fact all Ramsey graphs must obey certain "richness" properties characteristic of random graphs. More than 25 years ago, Erdős, Faudree and Sós conjectured that in any C-Ramsey graph there are $Ω\left(n^{5/2}\right)$ induced subgraphs, no pair of which have the same numbers of vertices and edges. Improving on earlier results of Alon, Balogh, Kostochka and Samotij, in this paper we prove this conjecture.

math.CO↗

arXiv2021

Friendly bisections of random graphs

Resolving a conjecture of Füredi from 1988, we prove that with high probability, the random graph $G(n,1/2)$ admits a friendly bisection of its vertex set, i.e., a partition of its vertex set into two parts whose sizes differ by at most one in which $n-o(n)$ vertices have at least as many neighbours in their own part as across. The engine of our proof is a new method to study stochastic processes driven by degree information in random graphs; this involves combining enumeration techniques with an abstract second moment argument.

math.CO↗

arXiv2021

List-decodability with large radius for Reed-Solomon codes

List-decodability of Reed-Solomon codes has received a lot of attention, but the best-possible dependence between the parameters is still not well-understood. In this work, we focus on the case where the list-decoding radius is of the form $r=1-\varepsilon$ for $\varepsilon$ tending to zero. Our main result states that there exist Reed-Solomon codes with rate $Ω(\varepsilon)$ which are $(1-\varepsilon, O(1/\varepsilon))$-list-decodable, meaning that any Hamming ball of radius $1-\varepsilon$ contains at most $O(1/\varepsilon)$ codewords. This trade-off between rate and list-decoding radius is best-possible for any code with list size less than exponential in the block length. By achieving this trade-off between rate and list-decoding radius we improve a recent result of Guo, Li, Shangguan, Tamo, and Wootters, and resolve the main motivating question of their work. Moreover, while their result requires the field to be exponentially large in the block length, we only need the field size to be polynomially large (and in fact, almost-linear suffices). We deduce our main result from a more general theorem, in which we prove good list-decodability properties of random puncturings of any given code with very large distance.

cs.IT↗

arXiv2021

Singularity of sparse random matrices: simple proofs

Consider a random $n\times n$ zero-one matrix with "density" $p$, sampled according to one of the following two models: either every entry is independently taken to be one with probability $p$ (the "Bernoulli" model), or each row is independently uniformly sampled from the set of all length-$n$ zero-one vectors with exactly $pn$ ones (the "combinatorial" model). We give simple proofs of the (essentially best-possible) fact that in both models, if $\min(p,1-p)\geq (1+\varepsilon)\log n/n$ for any constant $\varepsilon>0$, then our random matrix is nonsingular with probability $1-o(1)$. In the Bernoulli model this fact was already well-known, but in the combinatorial model this resolves a conjecture of Aigner-Horev and Person.

math.CO↗

arXiv2021

Lower bounds for superpatterns and universal sequences

A permutation $σ\in S_n$ is said to be $k$-universal or a $k$-superpattern if for every $π\in S_k$, there is a subsequence of $σ$ that is order-isomorphic to $π$. A simple counting argument shows that $σ$ can be a $k$-superpattern only if $n\ge (1/e^2+o(1))k^2$, and Arratia conjectured that this lower bound is best-possible. Disproving Arratia's conjecture, we improve the trivial bound by a small constant factor. We accomplish this by designing an efficient encoding scheme for the patterns that appear in $σ$. This approach is quite flexible and is applicable to other universality-type problems; for example, we also improve a bound by Engen and Vatter on a problem concerning $(k+1)$-ary sequences which contain all $k$-permutations.

math.CO↗

arXiv2020

Anticoncentration for subgraph counts in random graphs

Fix a graph $H$ and some $p\in (0,1)$, and let $X_H$ be the number of copies of $H$ in a random graph $G(n,p)$. Random variables of this form have been intensively studied since the foundational work of Erdős and Rényi. There has been a great deal of progress over the years on the large-scale behaviour of $X_H$, but the more challenging problem of understanding the small-ball probabilities has remained poorly understood until now. More precisely, how likely can it be that $X_H$ falls in some small interval or is equal to some particular value? In this paper we prove the almost-optimal result that if $H$ is connected then for any $x\in \mathbb{N}$ we have $\Pr(X_H=x)\le n^{1-v(H)+o(1)}$. Our proof proceeds by iteratively breaking $X_H$ into different components which fluctuate at "different scales", and relies on a new anticoncentration inequality for random vectors that behave "almost linearly".

math.CO↗

arXiv2020

Combinatorial anti-concentration inequalities, with applications

We prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some "Poisson-type" anti-concentration theorems that give bounds of the form 1/e + o(1) for the point probabilities of certain polynomials. Second, we prove an anti-concentration inequality for polynomials with nonnegative coefficients which extends the classical Erdős-Littlewood-Offord theorem and improves a theorem of Meka, Nguyen and Vu for polynomials of this type. As an application, we prove some new anti-concentration bounds for subgraph counts in random graphs.

math.CO↗

arXiv2020

An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs

Consider a quadratic polynomial $f\left(ξ_{1},\dots,ξ_{n}\right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem, which asks the same question for linear polynomials. As in the linear case, it is known that the point probabilities of $f$ can be as large as about $1/\sqrt{n}$, but still poorly understood is the "inverse" question of characterising the algebraic and arithmetic features $f$ must have if it has point probabilities comparable to this bound. In this paper we prove some results of an algebraic flavour, showing that if $f$ has point probabilities much larger than $1/n$ then it must be close to a quadratic form with low rank. We also give an application to Ramsey graphs, asymptotically answering a question of Kwan, Sudakov and Tran.

math.CO↗

arXiv2020

Almost all Steiner triple systems are almost resolvable

We show that for any n divisible by 3, almost all order-n Steiner triple systems admit a decomposition of almost all their triples into disjoint perfect matchings (that is, almost all Steiner triple systems are almost resolvable).

math.CO↗

arXiv2020

Almost all Steiner triple systems have perfect matchings

We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class). In fact, we prove a general upper bound on the number of perfect matchings in a Steiner triple system and show that almost all Steiner triple systems essentially attain this maximum. We accomplish this via a general theorem comparing a uniformly random Steiner triple system to the outcome of the triangle removal process, which we hope will be useful for other problems. Our methods can also be adapted to other types of designs; for example, we sketch a proof of the theorem that almost all Latin squares have transversals.

math.CO↗

arXiv2020

Halfway to Rota's basis conjecture

In 1989, Rota made the following conjecture. Given $n$ bases $B_{1},\dots,B_{n}$ in an $n$-dimensional vector space $V$, one can always find $n$ disjoint bases of $V$, each containing exactly one element from each $B_{i}$ (we call such bases transversal bases). Rota's basis conjecture remains wide open despite its apparent simplicity and the efforts of many researchers (for example, the conjecture was recently the subject of the collaborative "Polymath" project). In this paper we prove that one can always find $\left(1/2-o\left(1\right)\right)n$ disjoint transversal bases, improving on the previous best bound of $Ω\left(n/\log n\right)$. Our results also apply to the more general setting of matroids.

math.CO↗

arXiv2020

Universality of random permutations

It is a classical fact that for any $\varepsilon > 0$, a random permutation of length $n = (1 + \varepsilon) k^2 / 4$ typically contains a monotone subsequence of length $k$. As a far-reaching generalization, Alon conjectured that a random permutation of this same length $n$ is typically $k$-universal, meaning that it simultaneously contains every pattern of length $k$. He also made the simple observation that for $n = O(k^2 \log k)$, a random length-$n$ permutation is typically $k$-universal. We make the first significant progress towards Alon's conjecture by showing that $n = 2000 k^2 \log \log k$ suffices.

math.CO↗

arXiv2019

Nearly-linear monotone paths in edge-ordered graphs

How long a monotone path can one always find in any edge-ordering of the complete graph $K_n$? This appealing question was first asked by Chvátal and Komlós in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was $n^{2/3-o(1)}$. In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length $n^{1-o(1)}$.

math.CO↗

arXiv2019

Dense induced bipartite subgraphs in triangle-free graphs

The problem of finding dense induced bipartite subgraphs in $H$-free graphs has a long history, and was posed 30 years ago by Erdős, Faudree, Pach and Spencer. In this paper, we obtain several results in this direction. First we prove that any $K_t$-free graph with minimum degree at least $d$ contains an induced bipartite subgraph of minimum degree at least $c_t \log d/\log \log d$, confirming (asymptotically) several conjectures by Esperet, Kang and Thomassé. Complementing this result, we further obtain optimal bounds for this problem in the case of dense triangle-free graphs, and we also answer a question of Erdős, Janson, Łuczak and Spencer.

math.CO↗

arXiv2019

Non-trivially intersecting multi-part families

We say a family of sets is intersecting if any two of its sets intersect, and we say it is trivially intersecting if there is an element which appears in every set of the family. In this paper we study the maximum size of a non-trivially intersecting family in a natural "multi-part" setting. Here the ground set is divided into parts, and one considers families of sets whose intersection with each part is of a prescribed size. Our work is motivated by classical results in the single-part setting due to Erdős, Ko and Rado, and Hilton and Milner, and by a theorem of Frankl concerning intersecting families in this multi-part setting. In the case where the part sizes are sufficiently large we determine the maximum size of a non-trivially intersecting multi-part family, disproving a conjecture of Alon and Katona.

math.CO↗

arXiv2019

Hypergraph cuts above the average

An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut of size $m/2 + Ω(\sqrt{m})$, and this is best possible. That is, there exist cuts which exceed the expected size of a random cut by some multiple of the standard deviation. We study analogues of this and related results in hypergraphs. First, we observe that similarly to graphs, every m-edge k-uniform hypergraph has an r-cut whose size is $Ω(\sqrt m)$ larger than the expected size of a random r-cut. Moreover, in the case where k=3 and r=2 this bound is best possible and is attained by Steiner triple systems. Surprisingly, for all other cases (that is, if $k \geq 4$ or $r \geq 3$), we show that every m-edge k-uniform hypergraph has an r-cut whose size is $Ω(m^{5/9})$ larger than the expected size of a random r-cut. This is a significant difference in behaviour, since the amount by which the size of the largest cut exceeds the expected size of a random cut is now considerably larger than the standard deviation.

math.CO↗

arXiv2018

Anticoncentration for subgraph statistics

Consider integers $k,\ell$ such that $0\le \ell \le \binom{k}2$. Given a large graph $G$, what is the fraction of $k$-vertex subsets of $G$ which span exactly $\ell$ edges? When $G$ is empty or complete, and $\ell$ is zero or $\binom{k}{2}$, this fraction can be exactly 1. On the other hand, if $\ell$ is far from these extreme values, one might expect that this fraction is substantially smaller than 1. This was recently proved by Alon, Hefetz, Krivelevich and Tyomkyn who intiated the systematic study of this question and proposed several natural conjectures. Let $\ell^{*}=\min\{\ell,\binom{k}{2}-\ell\}$. Our main result is that for any $k$ and $\ell$, the fraction of $k$-vertex subsets that span $\ell$ edges is at most $\log^{O\left(1\right)}\left(\ell^{*}/k\right)\sqrt{k/\ell^{*}}$, which is best-possible up to the logarithmic factor. This improves on multiple results of Alon, Hefetz, Krivelevich and Tyomkyn, and resolves one of their conjectures. In addition, we also make some first steps towards some analogous questions for hypergraphs. Our proofs involve some Ramsey-type arguments, and a number of different probabilistic tools, such as polynomial anticoncentration inequalities, hypercontractivity, and a coupling trick for random variables defined on a "slice" of the Boolean hypercube.

math.CO↗