1-color-avoiding paths, special tournaments, and incidence geometry

Abstract

We discuss two approaches to a recent question of Loh: must a 3-colored transitive tournament on $N$ vertices have a 1-color-\emph{avoiding} path of vertex-length at least $N^{2/3}$? This question generalizes the Erd\H{o}s--Szekeres theorem on monotone subsequences. First, we define three canonical transformations on these tournaments called Color, Record, and Dual. We use these to establish a reduction to special tournaments with natural geometric and combinatorial properties. In many cases (including all known tight examples), these tournaments have recursive Gallai decompositions. Not all relevant tournaments have Gallai decompositions, but those that do satisfy the desired $N^{2/3}$ bound by recent work of Wagner, roughly analogous to earlier work of Fox, Grinshpun, and Pach on a similar \emph{undirected} problem. Second, we consider the related geometric problem of bounding \emph{slice-increasing} sets $S\subseteq [n]^3$, which---under an additional ordering hypothesis on $S$---was shown by Loh to be equivalent to the original question. In particular, we establish a rigorous connection from a problem of Szab\'o and Tardos, raise a stronger $L^2$-question on slice-counts, and mention a surprising overlap with the joints problem.