arXiv subjects
Publications and source records attributed to Julian Sahasrabudhe.
We show that the Erdős-Rogers function $f_{s,s+1}(n)$ satisfies $$f_{s,s+1}(n) = Θ( \sqrt{n \log n} )$$ for every $s \ge 2$. More precisely, we construct a $K_{s+1}$-free graph on $n$ vertices in which every set of at least $C(s)\sqrt{n \log n}$ vertices contains a copy of $K_s$ for some constant $C(s)$, which implies the upper bound. The matching lower bound follows from a theorem of Joret, Micek, Reed and Smid on the clique chromatic number of a graph.
We give a polynomial improvement to the cycle-complete Ramsey numbers \[ r(C_{\ell},K_k) \geq k^{1+1/(\ell- 2) + \varepsilon_{\ell} + o(1)}, \] for all fixed odd $\ell > 7$ with $k \rightarrow \infty$, for some $\varepsilon_{\ell} > 0$.
In this paper, we determine the $r$-colour size Ramsey number of the path $P_k$, up to constants. In particular, for every fixed $r \geq 2$ and $k \geq 100\log r$, we have \[ \widehat{R}_r(P_k)=Θ((r^2 \log r) \, k).\] Perhaps surprisingly, we do this by improving the lower bound on $\widehat{R}_r(P_k)$.
The $r$-colour Ramsey number $R_r(k)$ is the minimum $n \in \mathbb{N}$ such that every $r$-colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove, for each fixed $r \geqslant 2$, that $$R_r(k) \leqslant e^{-δk} r^{rk}$$ for some constant $δ= δ(r) > 0$ and all sufficiently large $k \in \mathbb{N}$. For each $r \geqslant 3$, this is the first exponential improvement over the upper bound of Erdős and Szekeres from 1935. In the case $r = 2$, it gives a different (and significantly shorter) proof of a recent result of Campos, Griffiths, Morris and Sahasrabudhe.
In many applications of the probabilistic method, one looks to study phenomena that occur ``with high probability''. More recently however, in an attempt to understand some of the most fundamental problems in combinatorics, researchers have been diving deeper into these probability spaces and understanding phenomena that occur at much smaller probability scales. Here I will survey a few of these ideas from the perspective of my own work in the area.
The Ramsey number $R(k)$ is the minimum $n \in \mathbb{N}$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that \[ R(k) \leqslant (4 - \varepsilon)^k \] for some constant $\varepsilon > 0$. This is the first exponential improvement over the upper bound of Erdős and Szekeres, proved in 1935.
Let $A$ be an $n\times n$ matrix with iid entries where $A_{ij} \sim \mathrm{Ber}(p)$ is a Bernoulli random variable with parameter $p = d/n$. We show that the empirical measure of the eigenvalues converges, in probability, to a deterministic distribution as $n \rightarrow \infty$. This essentially resolves a long line of work to determine the spectral laws of iid matrices and is the first known example for non-Hermitian random matrices at this level of sparsity.
We prove a new lower bound for the off-diagonal Ramsey numbers, \[ R(3,k) \geq \bigg( \frac{1}{3}+ o(1) \bigg) \frac{k^2}{\log k }\, , \] thereby narrowing the gap between the upper and lower bounds to a factor of $3+o(1)$. This improves the best known lower bound of $(1/4+o(1))k^2/\log k$ due, independently, to Bohman and Keevash, and Fiz Pontiveros, Griffiths and Morris, resulting from their celebrated analysis of the triangle-free process. As a consequence, we disprove a conjecture of Fiz Pontiveros, Griffiths and Morris that the constant $1/4$ is sharp.
Let $A_n$ be an $n\times n$ matrix with iid entries distributed as Bernoulli random variables with parameter $p = p_n$. Rudelson and Tikhomirov, in a beautiful and celebrated paper, show that the distribution of eigenvalues of $A_n \cdot (pn)^{-1/2}$ is approximately uniform on the unit disk as $n\rightarrow \infty$ as long as $pn \rightarrow \infty$, which is the natural necessary condition. In this paper we give a much simpler proof of this result, in its full generality, using a perspective we developed in our recent proof of the existence of the limiting spectral law when $pn$ is bounded. One feature of our proof is that it avoids the use of $ε$-nets entirely and, instead, proceeds by studying the evolution of the singular values of the shifted matrices $A_n-zI$ as we incrementally expose the randomness in the matrix.
Let $M$ be an $n\times n$ matrix with iid subgaussian entries with mean $0$ and variance $1$ and let $σ_n(M)$ denote the least singular value of $M$. We prove that \[\mathbb{P}\big( σ_{n}(M) \leq \varepsilon n^{-1/2} \big) = (1+o(1)) \varepsilon + e^{-Ω(n)}\] for all $0 \leq \varepsilon \ll 1$. This resolves, up to a $1+o(1)$ factor, a seminal conjecture of Spielman and Teng.
We show there exists a packing of identical spheres in $\mathbb{R}^d$ with density at least \[ (1-o(1))\frac{d \log d}{2^{d+1}}\, , \] as $d\to\infty$. This improves upon previous bounds for general $d$ by a factor of order $\log d$ and is the first asymptotically growing improvement to Rogers' bound from 1947.
Let $A$ be a $n \times n$ symmetric matrix with $(A_{i,j})_{i\leq j} $, independent and identically distributed according to a subgaussian distribution. We show that $$\mathbb{P}(σ_{\min}(A) \leq \varepsilon/\sqrt{n}) \leq C \varepsilon + e^{-cn},$$ where $σ_{\min}(A)$ denotes the least singular value of $A$ and the constants $C,c>0 $ depend only on the distribution of the entries of $A$. This result confirms a folklore conjecture on the lower-tail asymptotics of the least singular value of random symmetric matrices and is best possible up to the dependence of the constants on the distribution of $A_{i,j}$. Along the way, we prove that the probability $A$ has a repeated eigenvalue is $e^{-Ω(n)}$, thus confirming a conjecture of Nguyen, Tao and Vu.
Introduced by Erdős in 1950, a covering system of the integers is a finite collection of arithmetic progressions whose union is the set $\mathbb{Z}$. Many beautiful questions and conjectures about covering systems have been posed over the past several decades, but until recently little was known about their properties. Most famously, the so-called minimum modulus problem of Erdős was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most $10^{16}$. In this paper we answer another question of Erdős, asked in 1952, on the number of minimal covering systems. More precisely, we show that the number of minimal covering systems with exactly $n$ elements is \[ \exp\left( \left(\frac{4\sqrtτ}{3} + o(1)\right) \frac{n^{3/2}}{(\log n)^{1/2}} \right) \] as $n \to \infty$, where \[ τ= \sum_{t = 1}^\infty \left( \log \frac{t+1}{t} \right)^2. \] En route to this counting result, we obtain a structural description of all covering systems that are close to optimal in an appropriate sense.
A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of these objects was initiated by Erdős in 1950, and over the following decades he asked many questions about them. Most famously, he asked whether there exist covering systems with distinct moduli whose minimum modulus is arbitrarily large. This problem was resolved in 2015 by Hough, who showed that in any such system the minimum modulus is at most $10^{16}$. The purpose of this note is to give a gentle exposition of a simpler and stronger variant of Hough's method, which was recently used to answer several other questions about covering systems. We hope that this technique, which we call the distortion method, will have many further applications in other combinatorial settings.
This paper provides a connection between the concentration of a random variable and the distribution of the roots of its probability generating function. Let $X$ be a random variable taking values in $\{0,\ldots,n\}$ with $\mathbb{P}(X = 0)\mathbb{P}(X = n) > 0$ and with probability generating function $f_X$. We show that if all of the zeros $ζ$ of $f_X$ satisfy $|\arg(ζ)| \geq δ$ and $R^{-1} \leq |ζ| \leq R$ then \[ \operatorname{Var}(X) \geq c R^{-2π/δ}n, \] where $c > 0$ is a absolute constant. We show that this result is sharp, up to the factor $2$ in the exponent of $R$. As a consequence, we are able to deduce a Littlewood--Offord type theorem for random variables that are not necessarily sums of i.i.d.\ random variables.
Let $A$ be drawn uniformly at random from the set of all $n\times n$ symmetric matrices with entries in $\{-1,1\}$. We show that \[ \mathbb{P}( \det(A) = 0 ) \leq e^{-cn},\] where $c>0$ is an absolute constant, thereby resolving a well-known conjecture.
Let $f \in \mathbb{R}[x]$ be a polynomial with real coefficients. We say that $f$ is eventually non-negative if $f^m$ has non-negative coefficients for all sufficiently large $m \in \mathbb{N}$. In this short note, we give a classification of all eventually non-negative polynomials. This generalizes a theorem of De Angelis, and proves a conjecture of Bergweiler, Eremenko and Sokal
Let $M_n$ be drawn uniformly from all $\pm 1$ symmetric $n \times n$ matrices. We show that the probability that $M_n$ is singular is at most $\exp(-c(n\log n)^{1/2})$, which represents a natural barrier in recent approaches to this problem. In addition to improving on the best-known previous bound of Campos, Mattos, Morris and Morrison of $\exp(-c n^{1/2})$ on the singularity probability, our method is different and considerably simpler.