Related papers: Erd\H{o}s-Szekeres without induction

In this paper we completely resolve the well-known problem of Erd\H{o}s and Sauer from 1975 which asks for the maximum number of edges an $n$-vertex graph can have without containing a $k$-regular subgraph, for some fixed integer $k\geq 3$.…

Let $f_{s,k}(n)$ be the maximum possible number of $s$-term arithmetic progressions in a sequence $a_1<a_2<\ldots<a_n$ of $n$ integers which contains no $k$-term arithmetic progression. For all integers $k > s \geq 3$, we prove that…

A set $S$ of natural numbers is multiplicative Sidon if the products of all pairs in $S$ are distinct. Erd\H{o}s in 1938 studied the maximum size of a multiplicative Sidon subset of $\{1,\ldots, n\}$, which was later determined up to the…

We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4-uniform Ramsey number $r_4(5,n)$, and the same for the iterated $(k-4)$-fold logarithm of the $k$-uniform…

Let $P$ be a set of $n$ points in the plane. We consider a variation of the classical Erd\H{o}s-Szekeres problem, presenting efficient algorithms with $O(n^3)$ running time and $O(n^2)$ space complexity that compute: (1) A subset $S$ of $P$…

Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets $\mathcal{P}$, which are sets of $n$ points in the plane with pairwise integral distances where not all the…

We investigate Strichartz estimates for the nonlinear beam equation with initial data $f\in\dot{H}^s, g\in\dot{H}^{s-2}$ and $f\in H^s, g\in H^{s-2}$. We extend results of H. Lindblad and C. D.Sogge [10] and T. Cazenave and F. B. Weissler…

The Erd\H os Matching Conjecture states that the maximum size $f(n,k,s)$ of a family $\mathcal{F}\subseteq \binom{[n]}{k}$ that does not contain $s$ pairwise disjoint sets is $\max\{|\mathcal{A}_{k,s}|,|\mathcal{B}_{n,k,s}|\}$, where…

The Erd\"{o}s-Straus conjecture states that the equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ has positive integer solutions $x, y, z$ for every postive integers $n\ge 2$. We generalize the Erd\"{o}s-Straus equation, state…

Let $S$ be a set of $n$ points in general position in the plane, and let $X_{k,\ell}(S)$ be the number of convex $k$-gons with vertices in $S$ that have exactly $\ell$ points of $S$ in their interior. We prove several equalities for the…

Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph not containing a subhypergraph with $k$ edges on at most $s$ vertices. Recently, Delcourt and Postle, building on work of Glock, Joos, Kim,…

We study the extremal function $S^k_d(n)$, defined as the maximum number of regular $(k-1)$-simplices spanned by $n$ points in $\mathbb{R}^d$. For any fixed $d\geq2k\geq6$, we determine the asymptotic behavior of $S^k_d(n)$ up to a…

We extend the famous Erd\H{o}s-Szekeres theorem to $k$-flats in ${\mathbb{R}^d}$

Introduced by Erd\H{o}s 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…

Let $P$ be a set of points in general position in the plane. Join all pairs of points in $P$ with straight line segments. The number of segment-crossings in such a drawing, denoted by $\crg(P)$, is the \emph{rectilinear crossing number} of…

We consider a 2-player permutation game inspired by the celebrated Erd\H{o}s-Szekeres Theorem. The game depends on two positive integer parameters $a$ and $b$ and we determine the winner and give a winning strategy when $a \geq b$ and $b…

Stochastic convex optimization is one of the most well-studied models for learning in modern machine learning. Nevertheless, a central fundamental question in this setup remained unresolved: "How many data points must be observed so that…

In 1997, Erd\H{o}s asked whether for arbitrarily large $n$ there exists a set of $n$ points in $\mathbb{R}^2$ that determines $O(\frac{n}{\sqrt{\log n}})$ distinct distances while satisfying the local constraint that every 4-point subset…

Let n \geq l \geq 2 and q \geq 2. We consider the minimum N such that whenever we have N points in the plane in general position and the l-subsets of these points are colored with q colors, there is a subset S of n points all of whose…

If an $n$-uniform hypergraph can be 2-colored, then it is said to have property B. Erd\H{o}s (1963) was the first to give lower and upper bounds for the minimal size $m(n)$ of an $n$-uniform hypergraph without property B. His asymptotic…