相关论文: An improved lower bound for Erd\H{o}s--Szekeres pr…

Let f(n) denote the smallest positive integer such that every set of $f(n)$ points in general position in the Euclidean plane contains a convex n-gon. In a seminal paper published in 1935, Erd\H{o}s and Szekeres proved that f(n) exists and…

Propositions 1.1 -- 1.3 stated below contribute to results and certain problems considered in a paper by Erdos and Szekeres, on the behavior of products $\prod^n_1 (1-z^{a_j}), 1\leq a_1\leq \cdots\leq a_n$ integers. In the discussion,…

The classical Erd\H{o}s-Szekeres theorem dating back almost a hundred years states that any sequence of $(n-1)^2+1$ distinct real numbers contains a monotone subsequence of length $n$. This theorem has been generalised to higher dimensions…

Let $ES(n)$ be the minimal integer such that any set of $ES(n)$ points in the plane in general position contains $n$ points in convex position. The problem of estimating $ES(n)$ was first formulated by Erd\H{o}s and Szekeres, who proved…

For each $n$, let $\text{RD}(n)$ denote the minimum $d$ for which there exists a formula for the general polynomial of degree $n$ in algebraic functions of at most $d$ variables. In 1945, Segre called for a better understanding of the large…

In 1993, Fishburn and Graham established the following qualitative extension of the classical Erd\H{o}s-Szekeres theorem. If $N$ is sufficiently large with respect to $n$, then any $N\times N$ real matrix contains an $n\times n$ submatrix…

Erdos, Herzog and Piranian asked whether, for $n$ points in the plane with fixed diameter (maximum distance between points), an arrangement of a regular $n$-gon maximizes their product of all pairs of distances. Recently, it was discovered…

For $0\le k\le n$, write $\binom nk=uv$ where the primes dividing $u$ are at most $k$ and the primes dividing $v$ exceed $k$, and let $f(n)$ be the least $k$ with $u>n^2$; Erd\H{o}s problem 684 asks for bounds on $f(n)$. We resolve the…

Let $Z(N)$ denote the minimum number of zeros in $[0,2\pi]$ that a cosine polynomial of the form $$f_A(t)=\sum_{n\in A}\cos nt$$ can have when $A$ is a finite set of non-negative integers of size $|A|=N$. It is an old problem of Littlewood…

Let $ES(n)$ denote the minimum natural number such that every set of $ES(n)$ points in general position in the plane contains $n$ points in convex position. In 1935, Erd\H{o}s and Szekeres proved that $ES(n) \le {2n-4 \choose n-2}+1$. In…

A famous result by Erd\H{o}s and Szekeres (1935) asserts that, for all $k,d \in \mathbb{N}$, there is a smallest integer $n = g^{(d)}(k)$ such that every set of at least $n$ points in $\mathbb{R}^d$ in general position contains a $k$-gon,…

We prove new lower bounds on the maximum size of subsets $A\subseteq \{1,\dots,N\}$ or $A\subseteq \mathbb{F}_p^n$ not containing three-term arithmetic progressions. In the setting of $\{1,\dots,N\}$, this is the first improvement upon a…

In their seminal paper from 1983, Erd\H{o}s and Szemer\'edi showed that any $n$ distinct integers induce either $n^{1+\epsilon}$ distinct sums of pairs or that many distinct products, and conjectured a lower bound of $n^{2-o(1)}$. They…

For $2\leq s<t$, the Erd\H{o}s-Rogers function $f_{s,t}(n)$ measures how large a $K_s$-free induced subgraph there must be in a $K_t$-free graph on $n$ vertices. There has been an extensive amount of work towards estimating this function,…

Let $P_{<n}(z)$ be the Rudin-Shapiro polynomial of degree $n-1$. We show that $|P_{<n}(z)|\le \sqrt{6n-2}-1$ for all $n\ge0$ and $|z|=1$, confirming a longstanding conjecture. This bound is sharp in the case when $n=(2\cdot 4^k+1)/3$ and…

The Erd\H{o}s-Rogers function $f_{s,t}$ measures how large a $K_s$-free induced subgraph there must be in a $K_t$-free graph on $n$ vertices. While good estimates for $f_{s,t}$ are known for some pairs $(s,t)$, notably when $t=s+1$, in…

According to the Erd\H{o}s-Szekeres theorem, for every $n$, a sufficiently large set of points in general position in the plane contains $n$ in convex position. In this note we investigate the line version of this result, that is, we want…

In this paper we give a conditional improvement to the Elekes-Szab\'{o} problem over the rationals, assuming the Uniformity Conjecture. Our main result states that for $F\in \mathbb{Q}[x,y,z]$ belonging to a particular family of…

In 1974, Erd\H{o}s and Kleitman conjectured that if a family $\mathcal{F}\subseteq 2^{[n]}$ contains no matching of size \(s\) and is maximal with respect to this property, then $ |\mathcal{F}|\ge \left(1-2^{-(s-1)}\right)\cdot 2^{n}. $ For…

In this paper, we propose new lower and upper bounds on the linear extension complexity of regular $n$-gons. Our bounds are based on the equivalence between the computation of (i) an extended formulation of size $r$ of a polytope $P$, and…