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We provide a cyclic permutation analogue of the ErdH 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 cyclic sub-permutation of length $ell+1$, and show that the result is tight. We also characterize all maximum-length cyclic permutations that do not have an increasing cyclic sub-permutation of length $k+1$ or a decreasing cyclic sub-permutation of length $ell+1$.
We extend the famous ErdH{o}s-Szekeres theorem to $k$-flats in ${mathbb{R}^d}$
In 1935, ErdH{o}s and Szekeres proved that $(m-1)(k-1)+1$ is the minimum number of points in the plane which definitely contain an increasing subset of $m$ points or a decreasing subset of $k$ points (as ordered by their $x$-coordinates). We consider
A perfect matching of a complete graph $K_{2n}$ is a 1-regular subgraph that contains all the vertices. Two perfect matchings intersect if they share an edge. It is known that if $mathcal{F}$ is family of intersecting perfect matchings of $K_{2n}$, t
Given a sequence $mathbf{k} := (k_1,ldots,k_s)$ of natural numbers and a graph $G$, let $F(G;mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,dots,s$ such that, for every $c in {1,dots,s}$, the edges of colour $c$ conta
For a 2-connected graph $G$ on $n$ vertices and two vertices $x,yin V(G)$, we prove that there is an $(x,y)$-path of length at least $k$ if there are at least $frac{n-1}{2}$ vertices in $V(G)backslash {x,y}$ of degree at least $k$. This strengthens a