New results on the odd- and unique-Ramsey numbers

TL;DR

This paper advances understanding of odd- and unique-Ramsey numbers by establishing new bounds for specific graphs, analyzing host graph conditions, and highlighting gaps between these two types of numbers.

Contribution

It provides new asymptotic bounds for odd-Ramsey numbers of bipartite graphs, explores their behavior in super-Dirac host graphs, and compares them with unique-Ramsey numbers for cycles.

Findings

Established a lower bound for r_odd(n,K_{s,t}) when s is odd and t is even.

Proved non-trivial bounds for odd-Ramsey numbers of Hamilton cycles in graphs with high minimum degree.

Demonstrated a polynomial gap between r_odd(n,C_n) and r_u(n,C_n).

Abstract

The odd-Ramsey number $r_{\text{odd}}(n,H)$ of a graph $H$ is the minimum number of colors needed to edge-color $K_n$ so that in every copy of $H$ some color occurs an odd number of times, and the unique-Ramsey number $r_{\text{u}}(n,H)$ is the corresponding notion in which some color is required to occur not only an odd number of times but exactly once. In this paper, we address three questions from previous papers. We show $r_{\text{odd}}(n,K_{s,t})> n^{1/\left(\frac s2+\frac 1{2\lfloor t/8 \rfloor}\right)}$ when $s\leq t$ and $s$ is odd and $t$ is even, which is log-asymptotically tight when $s$ is fixed and $t\to\infty$. Next, we consider the odd-Ramsey number when the host graph to be edge-colored is a super-Dirac graph, and show that in any host graph with minimum degree at least $n/2+2$, the odd-Ramsey number of Hamilton cycles is non-trivial. Finally, we show that $r_\text{u}(n,C_n)> n/4$, which leads to a polynomial gap between $r_\text{odd}(n,C_n)$ and $r_\text{u}(n,C_n)$.