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

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…

In 1959, Erd\H{o}s and Szekeres posed a series of problems concerning the size of polynomials of the form $$ P_n(z) = \prod_{j=1}^n (1 - z^{s_j}), $$ where $s_1, \dots, s_n$ are positive integers. Of particular interest is the quantity $$…

The Erd\H{o}s-Szekeres Theorem stated in terms of graphs says that any red-blue coloring of the edges of the ordered complete graph $K_{rs+1}$ contains a red copy of the monotone increasing path with $r$ edges or a blue copy of the monotone…

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 a seminal paper in 1959, Marcus and Ree proved that every $n\times n$ bistochastic matrix $A$ satisfies $\|A\|_{\operatorname{F}}^2\leq \max_{\sigma\in S_n}A_{i,\sigma(i)}$ where $S_n$ is the symmetric group on $\{1, \ldots, n\}$.…

A famous conjecture of Erd\H{o}s asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem…

In 1965 Erd\H os conjectured that for all $k\ge2$, $s\ge1$ and $n\ge k(s+1)$, an $n$-vertex $k$-uniform hypergraph $\F$ with $\nu(\F)=s$ cannot have more than \newline $\max\{\binom{sk+k-1}k,\;\binom nk-\binom{n-s}k\}$ edges. It took almost…

In 1964, Erd\H{o}s, Hajnal and Moon introduced a saturation version of Tur\'an's classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that…

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…

For a positive integer $k$, a graph property $\mathcal{H}$, and a graph parameter $\mathcal{P}$, let $\operatorname{ex}_{\mathcal{P}}(n, \mathcal{H}; \delta \geq k)$ denote the maximum value of $\mathcal{P}$ over all $n$-vertex graphs with…

Let $n, d$ be integers with $1 \leq d \leq \left \lfloor \frac{n-1}{2} \right \rfloor$, and set $h(n,d):={n-d \choose 2} + d^2$. Erd\H{o}s proved that when $n \geq 6d$, each nonhamiltonian graph $G$ on $n$ vertices with minimum degree…

Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ edges on at most $s$ vertices. Brown, Erd\H{o}s and S\'os conjectured in 1973 that the limit $\lim_{n\rightarrow…

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…

The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent set of size $n$. A well-known conjecture of Erd\H{o}s and Hajnal states that for any fixed…

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…

Let $n, d$ be integers with $1 \leq d \leq \left \lfloor \frac{n-1}{2} \right \rfloor$, and set $h(n,d):={n-d \choose 2} + d^2$ and $e(n,d):= \max\{h(n,d),h(n, \left \lfloor \frac{n-1}{2} \right \rfloor)\}$. Because $h(n,d)$ is quadratic in…

The classical Ramsey numbers $r(s,t)$ denote the minimum $n$ such that every red-blue coloring of the edges of the complete graph $K_n$ contains either a red clique of order $s$ or a blue clique of order $t$. These quantities are the…

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…

Let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an $r$-graph on $n$ vertices in which every $k$ edges span more than $s$ vertices. Brown, Erd\H{o}s and S\'{o}s in 1973 conjectured that for every $k\geq 2$, the limit…

In 1965 Erd\H{o}s conjectured that the number of edges in k-uniform hypergraphs on n vertices in which the largest matching has s edges is maximized for hypergraphs of one of two special types. We settled this conjecture in the affirmative…