Color avoidance for monotone paths
Eion Mulrenin, Cosmin Pohoata, Dmitrii Zakharov
TL;DR
This work analyzes color-avoiding Ramsey numbers for tight monotone paths in k-uniform hypergraphs, introducing A_k(n; r, s) as the threshold where an r-coloring yields a tight monotone path of length n using at most s colors. It develops two main advances: first, for uniformity k = 3 with s > r/2, A_3(n; r, s) is polynomial in n (e.g., A_3(n; 3, 2) ≤ n^9), and second, for general k ≥ 4, the tower height in bounds drops by one exponential compared to the standard MS bounds, yielding A_k(n; r, s) ≤ T_{k-3}(n^{C binom{r}{s}^2 log binom{r}{s}}). The key techniques blend Seidenberg-type down-set encodings, majority-tournament domination theory, and a generalized S-increasing-sequences framework to relate color-avoiding problems to more tractable combinatorial objects. These results refine our understanding of how color-avoidance alters the complexity of Ramsey-type questions and provide a versatile toolkit for analyzing color-avoiding monotone paths across uniformities. The work also establishes a characterization of A_k(n; r, s) via m_k(n; binom{r}{s}, binom{r-1}{s-1}) and introduces generalized S-increasing sequences as a unifying concept for these problems.
Abstract
In 2014, Moshkovitz and Shapira determined the tower height for hypergraph Ramsey numbers of tight monotone paths. We address the color-avoiding version of this problem in which one no longer necessarily seeks a monochromatic subgraph, but rather one which avoids some colors. This problem was previously studied in uniformity two by Loh and by Gowers and Long. We show, in general, that the tower height for such Ramsey numbers requires one less exponential than in the usual setting. The transition occurs at uniformity three, where the usual Ramsey numbers of monotone paths of length $n$ are exponential in $n$, but the color-avoiding Ramsey numbers turn out to be polynomial.