Ramsey numbers of K_s + mK_t versus K_n

TL;DR

This paper establishes an upper bound on the Ramsey number for the join of a clique and multiple disjoint cliques versus a large clique, confirming a conjecture and matching known bounds in a special case.

Contribution

It proves a new asymptotic bound for Ramsey numbers involving joins of cliques and disjoint cliques, settling a previously open problem.

Findings

Established an O( n^{s+t-1} / (log n)^{s+t-2} ) bound for R(K_s + mK_t, K_n).

Confirmed the bound is tight up to a constant for the case (s,t)=(0,3).

Matched the classical Kim result for R(K_3, K_n).

Abstract

For integers m >= 1, s >= 0, and t >= 1, let K_s + mK_t denote the join of a clique K_s and m vertex-disjoint copies of K_t. We prove that for fixed m >= 1, t >= 1, and s >= 0, R(K_s + mK_t, K_n) = O( n^{s+t-1} / (log n)^{s+t-2} ). This settles a problem proposed by Liu and Li (2026). Moreover, for (s,t) = (0,3) the bound is tight up to a constant factor, matching the classical result R(K_3, K_n) = Theta( n^2 / log n ) of Kim (1995).