Ramsey numbers of graphs with most degrees bounded in random graphs

TL;DR

This paper investigates the thresholds for Ramsey properties of random graphs with bounded degrees, establishing conditions under which certain monochromatic subgraphs almost surely appear or do not appear.

Contribution

It introduces new threshold bounds for Ramsey properties in random graphs with degree constraints, connecting these thresholds to classical Ramsey numbers.

Findings

Identifies upper and lower threshold functions for Ramsey properties in random graphs.

Establishes the relation between degree bounds and Ramsey thresholds in ${ m G}(N,p)$.

Defines weak Ramsey thresholds via edge coloring of random graphs.

Abstract

For graphs $F$ and $G$, let $F\to G$ signify that any red/blue edge coloring of $F$ contains a monochromatic $G$. Denote by ${\cal G}(N,p)$ the random graph space of order $N$ and edge probability $p$. Using the regularity method, one can show that for any fixed $p\in (0,1]$, almost all graphs $F\in{\cal G}(cn,p)$ have $F\to G$ for any graph $G$ of order $n$ and all but at most $m$ degrees bounded, where $c$ is an integer depending on $p$ and $m$. Note that $r(K_{m,n})\sim 2^m n$ and $r(K_m+\overline{K}_n)\sim 2^m n$ as $n\to\infty$, for which we investigate the relation between $c$ and $p$. Let $N=\lfloor c\,2^m n\rfloor$ with $c>1$ and $p_u,p_\ell=\frac{1}{c^{1/m}}(1\pm \sqrt{\frac{M\log n}{n}}\,)$, where $M=M(c,m)>0$. It is shown that $p_u$ and $p_\ell$ are Ramsey thresholds of $K_{m,n}$ in ${\cal G}(N,p)$. Namely, almost all $F\in{\cal G}(N,p_u)$ and almost no $F\in{\cal G}(N,p_\ell)$ have $F\to K_{m,n}$, respectively. Moreover, it is shown that $p_u$ and $p_\ell$ are (ordinary) upper threshold and lower threshold of $K_m+\overline{K}_n$ to appear in ${\cal G}(N,p/2)$, respectively. We show that ${\cal G}(N,p/2)$ can be identified as the set of red (or blue) graphs obtained from $F\in{\cal G}(N,p)$ by red/blue edge coloring of $F$ with probability $1/2$ for each color, which leads to the definition of the weak Ramsey thresholds.