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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 $ell+1$.
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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 their result from an on-line game perspective: Let points be determined one by one by player A first determining the $x$-coordinate and then player B determining the $y$-coordinate. What is the minimum number of points such that player A can force an increasing subset of $m$ points or a decreasing subset of $k$ points? We introduce this as the ErdH{o}s-Szekeres on-line number and denote it by $text{ESO}(m,k)$. We observe that $text{ESO}(m,k) < (m-1)(k-1)+1$ for $m,k ge 3$, provide a general lower bound for $text{ESO}(m,k)$, and determine $text{ESO}(m,3)$ up to an additive constant.
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 $K_{2n}$, then $|mathcal{F}| leq (2(n-1) - 1)!!$ and if equality holds, then $mathcal{F} = mathcal{F}_{ij}$ where $ mathcal{F}_{ij}$ is the family of all perfect matchings of $K_{2n}$ that contain some fixed edge $ij$. We give a short algebraic proof of this result, resolving a question of Godsil and Meagher. Along the way, we show that if a family $mathcal{F}$ is non-Hamiltonian, that is, $m cup m ot cong C_{2n}$ for any $m,m in mathcal{F}$, then $|mathcal{F}| leq (2(n-1) - 1)!!$ and this bound is met with equality if and only if $mathcal{F} = mathcal{F}_{ij}$. Our results make ample use of a somewhat understudied symmetric commutative association scheme arising from the Gelfand pair $(S_{2n},S_2 wr S_n)$. We give an exposition of a few new interesting objects that live in this scheme as they pertain to our results.
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 $c$ contain no clique of order $k_c$. Write $F(n;mathbf{k})$ to denote the maximum of $F(G;mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by ErdH{o}s and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases. In previous work with Yilma, we constructed a finite optimisation problem whose maximum is equal to the limit of $log_2 F(n;mathbf{k})/{nchoose 2}$ as $n$ tends to infinity and proved a stability theorem for complete multipartite graphs $G$. In this paper we provide a sufficient condition on $mathbf{k}$ which guarantees a general stability theorem for any graph $G$, describing the asymptotic structure of $G$ on $n$ vertices with $F(G;mathbf{k}) = F(n;mathbf{k}) cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem. We apply our theorem to systematically recover existing stability results as well as all cases with $s=2$. The proof uses a novel version of symmetrisation on edge-coloured weighted multigraphs.
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 well-known theorem due to ErdH{o}s and Gallai in 1959. As the first application of this result, we show that a 2-connected graph with $n$ vertices contains a cycle of length at least $2k$ if it has at least $frac{n}{2}+k$ vertices of degree at least $k$. This confirms a 1975 conjecture made by Woodall. As another applications, we obtain some results which generalize previous theorems of Dirac, ErdH{o}s-Gallai, Bondy, and Fujisawa et al., present short proofs of the path case of Loebl-Koml{o}s-S{o}s Conjecture which was verified by Bazgan et al. and of a conjecture of Bondy on longest cycles (for large graphs) which was confirmed by Fraisse and Fournier, and make progress on a conjecture of Bermond.
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