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

TL;DR

This paper investigates a combinatorial problem related to 3-colored transitive tournaments and geometric configurations, establishing reductions, bounds, and connections to other geometric problems, advancing understanding of color-avoiding paths.

Contribution

It introduces canonical transformations and reductions to special tournaments, proving bounds for certain configurations and connecting combinatorial and geometric problems.

Findings

Tournaments with Gallai decompositions satisfy the N^{2/3} bound.

Connections established between tournament problems and slice-increasing sets.

Identifies overlaps with the joints problem in 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.