Constructive Ramsey Numbers for Loose Hyperpaths

Abstract

For positive integers k and $\ell$, a k-uniform hypergraph is called a loose path of length $\ell$, and denoted by $P_\ell ^{(k)}$, if its vertex set is $\{v_1, v_2, \ldots , v_{(k-1)\ell +1}\}$ and the edge set is $\{e_i = \{ v_{(i-1)(k-1)+q} : 1 \le q \le k \},\ i=1,\dots ,\ell \}$, that is, each pair of consecutive edges intersects on a single vertex. Let $R(P_\ell ^{(k)};r)$ be the multicolor Ramsey number of a loose path that is the minimum n such that every r-edge-coloring of the complete k-uniform hypergraph $K_n^{(k)}$ yields a monochromatic copy of $P_\ell ^{(k)}$. In this note we are interested in constructive upper bounds on $R(P_\ell ^{(k)};r)$ which means that on the cost of possibly enlarging the order of the complete hypergraph, we would like to efficiently find a monochromatic copy of $P_\ell ^{(k)}$ in every coloring. In particular, we show that there is a constant $c>0$ such that for all $k\ge 2$, $\ell \ge 3$, $2\le r\le k-1$, and $n\ge k(\ell +1)r(1+\ln (r))$, there is an algorithm such that for every r-edge-coloring of the edges of $K_n^{(k)}$, it finds a monochromatic copy of $P_\ell ^{(k)}$ in time at most $cn^k$.

A. Dudek—supported in part by Simons Foundation Grant #522400.

A. Ruciński—supported in part by the Polish NSC grant 2014/15/B/ST1/01688.