Fan-complete Ramsey numbers

Abstract

For graphs G and H, we consider Ramsey numbers $r(G,H)$ with tight lower bounds, namely, $r(G,H)\ge \left(\chi \right(G)-1)\left(\right|H|-1)+1$, where $\chi \left(G\right)$ denotes the chromatic number of G and $\left|H\right|$ denotes the number of vertices in H. We say H is G-good if the equality holds.

Let $G+H$ be the join graph obtained from graphs G and H by adding all edges between the disjoint vertex sets of G and H. Let nH denote the union graph of n disjoint copies of H. We show that $K_1+nH$ is $K_p$-good if n is sufficiently large. In particular, the fan-graph $F_n=K_1+n{K}_2$ is $K_p$-good if $n=\Omega \left(p^2\right)$, improving previous tower-type lower bounds for n due to Li and Rousseau (1996). Moreover, we give a stronger lower bound inequality for Ramsey number $r(G,K_1+F)$ for the case of $G=K_p(a_1,a_2,\dots ,a_p)$, the complete p-partite graph with $a_1=1$ and $a_i\le a_{i+1}$. In particular, using a stability-supersaturation lemma by Fox, He and Wigderson (2023), we show that for any fixed graph H,$r(G,K_1+nH)=\{\begin{array}{c}(p-1)\left(n\right|H|+a_2-1)+1\\ \text{if }n\left|H\right|+a_2-1\text{is even}\\ \text{or }{a}_2-1\text{is even,}\\ (p-1)\left(n\right|H|+a_2-2)+1\text{otherwise,}\end{array}$ where $G=K_p(1,a_2,\dots ,a_p)$ with $a_i$'s satisfying some mild conditions and n is sufficiently large. The special case of $H=K_1$ gives an answer to Burr's question (1981) about the discrepancy of $r(G,K_{1,n})$ from G-goodness for sufficiently large n. All bounds of n we obtain are not of tower-types.