Generalized Ramsey Numbers in the Hypercube

TL;DR

This paper investigates the minimum number of colors needed to edge-color hypercubes so that all cycles of a certain length contain a specified minimum number of colors, providing new bounds for various parameters.

Contribution

It establishes new asymptotic bounds for generalized Ramsey numbers in hypercubes for specific cycle lengths and color constraints, extending previous research.

Findings

For k ≥ 6 and 3 ≤ q ≤ k/2+1, f(Q_n, C_k, q) = o(n^{(k/2-1)/(k-q+1)}).

Derived upper and lower bounds for special cases k=4 and k=6.

Extended understanding of coloring properties in hypercube graphs related to cycle colorings.

Abstract

We study the generalized Ramsey numbers $f(Q_n, C_{k}, q)$, that is, the minimum number of colors needed to edge-color the hypercube $Q_n$ so that every copy of the cycle $C_{k}$ has at least $q$ colors. Our main result is that for any integers $k,q$ satisfying $k \geq 6$ and $3 \leq q \leq k/2+1$, we have $f(Q_n, C_{k}, q)= o\left( n^{\frac{k/2-1}{k-q+1}} \right).$ We also prove a few other upper and lower bounds in the special cases $k=4$ and $k=6$. This continues the line of research initiated by Faudree, Gy\'arf\'as, Lesniak, and Schelp and Mubayi and Stading who studied the case $k=q$, and by Conder who considered the case $k=6$ and $q=2$.