On-line size Ramsey number for monotone k-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 number Rt (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 path P (k) r