On-line size Ramsey number for monotone [formula omitted]-uniform ordered paths with uniform looseness

Abstract

An ordered hypergraph is a hypergraph H with a specified linear ordering of the vertices, and the appearance of an ordered hypergraph G in H must respect the specified order on V(G). In on-line Ramsey theory, Builder iteratively presents edges that Painter must immediately color. The t-color on-line size Ramsey numberR̃t(G) of an ordered hypergraph G is the minimum number of edges Builder needs to play (on a large ordered set of vertices) to force Painter using t colors to produce a monochromatic copy of G. The monotone tight pathPr(k) is the ordered hypergraph with r vertices whose edges are all sets of k consecutive vertices.We obtain good bounds on R̃t(Pr(k)). Letting m=r−k+1 (the number of edges in Pr(k)), we prove mt−1∕(3t)≤R̃t(Pr(2))≤tmt+1. For general k, a trivial upper bound is Rk, where R is the least number of vertices in a k-uniform (ordered) hypergraph whose t-colorings all contain Pr(k) (and is a tower of height k−2). We prove R∕(klgR)≤R̃t(Pr(k))≤R(lgR)2+ε, where ε is a positive constant and tm is sufficiently large in terms of ε−1. Our upper bounds improve prior results when t grows faster than m∕logm. We also generalize our results to ℓ-loose monotone paths, where each successive edge begins ℓ vertices after the previous edge.