Related papers: Exponential Erd\H{o}s-Szekeres theorem for matrice…

In this short note, we extend the celebrated results of Tao and Vu, and Krishnapur on the universality of empirical spectral distributions to a wide class of inhomogeneous complex random matrices, by showing that a technical and…

We prove the 4-uniform Erd\H{o}s Matching Conjecture for every matching number $s\ge 6961$. The proof has two parts. First, building on ideas from Frankl--R\"odl--Ruci\'nski, we formulate a general finite-board criterion for the $r$-uniform…

We provide a cyclic permutation analogue of the Erd\H os-Szekeres theorem. In particular, we show that every cyclic permutation of length $(k-1)(\ell-1)+2$ has either an increasing cyclic sub-permutation of length $k+1$ or a decreasing…

Burkill and Mirsky, and Kalmanson, prove independently that, for every $r\ge 2, n\ge 1$, there is a sequence of $r^{2^n}$ vectors in $\mathbb R^n$, which does not contain a subsequence of $r+1$ vectors $v^1, v^2,\dots,v^{r+1}$ such that,…

Let $F\in\mathbb{C}[x,y,s,t]$ be an irreducible constant-degree polynomial, and let $A,B,C,D\subset\mathbb{C}$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{8/3})$ points of the Cartesian product $A\times B\times…

In 1847, Kirkman proved that there exists a Steiner triple system on $n$ vertices (equivalently a triangle decomposition of the edges of $K_n$) whenever $n$ satisfies the necessary divisibility conditions (namely $n\equiv 1,3 \mod 6$). In…

A famous theorem of Erdos and Szekeres states that any sequence of $n$ distinct real numbers contains a monotone subsequence of length at least $\sqrt{n}$. Here, we prove a positive fraction version of this theorem. For $n > (k-1)^2$, any…

We show that if a simplicial complex is a near-cone of sufficiently high depth, then the only maximum families of small pairwise intersecting faces are those with a common intersection. Thus, near-cones of sufficiently high depth satisfy…

In 1990 Erd\H{o}s, Faudree, Rousseau and Schelp proved that for $k\geq 2$, every graph with $n\geq k+1$ vertices and $(k-1)(n-k+2)+\binom{k-2}{2}+1$ edges contains a subgraph of minimum degree $k$ on at most $n-\sqrt{n}/\sqrt{6k^3}$…

We show that for every integer $n\geq 1$ there exists a graph $G_n$ with $(1+o(1))n$ vertices and $n^{1 + o(1)}$ edges such that every $n$-vertex planar graph is isomorphic to a subgraph of $G_n$. The best previous bound on the number of…

The conjecture of Brown, Erd\H{o}s and S\'os from 1973 states that, for any $k \ge 3$, if a $3$-uniform hypergraph $H$ with $n$ vertices does not contain a set of $k+3$ vertices spanning at least $k$ edges then it has $o(n^2)$ edges. The…

This paper studies a problem of Erd\"{o}s concerning lattice cubes. Given an $N \times N \times N$ lattice cube, we want to find the maximum number of vertices one can select so that no eight corners of a rectangular box are chosen…

Let $ES_{d}(n)$ be the smallest integer such that any set of $ES_{d}(n)$ points in $\mathbb{R}^{d}$ in general position contains $n$ points in convex position. In 1960, Erd\H{o}s and Szekeres showed that $ES_{2}(n) \geq 2^{n-2} + 1$ holds,…

We give simple criteria to identify the exponential order of magnitude of the absolute value of the determinant for wide classes of random matrix models, not requiring the assumption of invariance. These include Gaussian matrices with…

An Erd\"os matrix $E$ is a bistochastic matrix whose sum of squares of entries (Frobenius norm squared) equals its maxtrace (maximum of all the $\sigma$-traces for permutations $\sigma$'s). We characterize all Erd\"os $E$ by the patterns of…

Kupavskii proved a codegree version of the Erd\H{o}s--Ko--Rado theorem by showing that for an intersecting family $\mathcal{F} \subseteq \binom{[n]}{k}$ with $n \geq 2k +3d/(1-d/k)$, the minimum $d$-degree of $\mathcal{F}$ is at most…

The Ramsey number $r(G)$ of a graph $G$ is the minimum $N$ such that every red-blue coloring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of $G$. Determining or estimating these numbers is one of the…

Planar graphs are known to allow subexponential algorithms running in time $2^{O(\sqrt n)}$ or $2^{O(\sqrt n \log n)}$ for most of the paradigmatic problems, while the brute-force time $2^{\Theta(n)}$ is very likely to be asymptotically…

This paper, originally written in Hungarian by D\'{e}nes K\H{o}nig in 1931, proves that in a bipartite graph, the minimum vertex cover and the maximum matching have the same size. This statement is now known as K\H{o}nig's theorem. The…

A folklore result attributed to P\'olya states that there are $(1 + o(1))2^{\binom{n}{2}}/n!$ non-isomorphic graphs on $n$ vertices. Given two graphs $G$ and $H$, we say that $G$ is a unique subgraph of $H$ if $H$ contains exactly one…