Monochromatic loose paths in multicolored $k$-uniform cliques

Abstract

For integers $k\ge 2$ and $\ell\ge 0$, a $k$-uniform hypergraph is called aloose path of length $\ell$, and denoted by $P_\ell^{(k)}$, if it consists of$\ell$ edges $e_1,\dots,e_\ell$ such that $|e_i\cap e_j|=1$ if $|i-j|=1$ and$e_i\cap e_j=\emptyset$ if $|i-j|\ge2$. In other words, each pair ofconsecutive edges intersects on a single vertex, while all other pairs aredisjoint. Let $R(P_\ell^{(k)};r)$ be the minimum integer $n$ such that every$r$-edge-coloring of the complete $k$-uniform hypergraph $K_n^{(k)}$ yields amonochromatic copy of $P_\ell^{(k)}$. In this paper we are mostly interested inconstructive upper bounds on $R(P_\ell^{(k)};r)$, meaning that on the cost ofpossibly enlarging the order of the complete hypergraph, we would like toefficiently find a monochromatic copy of $P_\ell^{(k)}$ in every coloring. Inparticular, we show that there is a constant $c>0$ such that for all $k\ge 2$,$\ell\ge3$, $2\le r\le k-1$, and $n\ge k(\ell+1)r(1+\ln(r))$, there is analgorithm such that for every $r$-edge-coloring of the edges of $K_n^{(k)}$, itfinds a monochromatic copy of $P_\ell^{(k)}$ in time at most $cn^k$. We alsoprove a non-constructive upper bound $R(P_\ell^{(k)};r)\le(k-1)\ell r$