Ramsey numbers of cliques versus monotone paths

Abstract

One formulation of the Erdos-Szekeres monotone subsequence theorem states that for any red/blue coloring of the edge set of the complete graph on $\{1, 2, \ldots, N\}$, there exists a monochromatic red $s$-clique or a monochromatic blue increasing path $P_n$ with $n$ vertices, provided $N >(s-1)(n-1)$. %We had previously shown that a suitable generalization of this problem to quadruple systems is essentially equivalent to classical diagonal hypergraph Ramsey numbers. Here, we prove a similar statement as above in the off-diagonal case for triple systems, with the quasipolynomial bound $N>2^{c(\log n)^{s-1}}$. For the $t$th power $P_n^t$ of the ordered increasing graph path with $n$ vertices, we prove a near linear bound $c\, n(\log n)^{s-2}$ which improves the previous bound that applied to a more general class of graphs than $P_n^t$ due to Conlon-Fox-Lee-Sudakov.