Induced Ramsey numbers for fans

TL;DR

This paper establishes quadratic upper bounds for induced Ramsey numbers involving fans and stars, advancing understanding of graph colorings that avoid certain induced subgraphs.

Contribution

It proves a quadratic upper bound for induced Ramsey numbers for fixed graphs and fans, and determines exact values for specific star-fan pairs.

Findings

Quadratic upper bound for fixed G and fan graphs F_n.

Exact induced Ramsey number for K_{1,2} and F_n is 3n+4.

Constructive coloring methods used to derive bounds.

Abstract

The induced Ramsey number $r_{\mathrm{ind}}(G,H)$ is defined as the minimum order of a graph $F$ on such that any 2-coloring of its edges with red and blue leads to either a red induced copy of $G$ or a blue induced copy of $H$. Motivated by the Kohayakawa-Pr\"omel-R\"odl conjecture, we prove that a quadratic upper bound $\mathrm{r}_{\text {ind}}\left(G, F_n\right) \leq C n^2$ for fixed $G$, where $F_n$ is a graph with one central vertex, $2n$ leaf vertices, and $n$ disjoint edges. In particular, for star graphs $K_{1, \ell}$ $(\ell \leq n)$, constructive coloring and matching arguments yield $2 n+2 \ell-1 \leq \mathrm{r}_{\text {ind}}\left(K_{1, \ell}, F_n\right) \leq(\ell+n-1)(\ell+1)+1$, with the exact value $\mathrm{r}_{\text {ind}}\left(K_{1,2}, F_n\right)=3 n+4$.