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Ramsey numbers of K_s + mK_t versus K_n

Lulu Dai, Qizhong Lin

TL;DR

The paper resolves the growth rate of the Ramsey number $R(K_s+mK_t, K_n)$ for fixed $m,t,s$ by establishing the upper bound $R(K_s+mK_t, K_n) = O\left( \frac{n^{s+t-1}}{(\log n)^{s+t-2}} \right)$. The authors prove this via induction on $s$, leveraging two key tools: the AKS triangle-counting lemma to convert sparse triangle structure into large independent sets, and the Li–Rouss bound to translate degree constraints into independence numbers. The result tightens previous work by eliminating a $\log\log n$ factor in the $t\ge 3$ regime and achieves tightness up to constants for $(s,t)=(0,3)$, aligning with Kim’s classical bound $R(K_3,K_n)=\Theta\left(\frac{n^2}{\log n}\right)$. The approach generalizes known bounds for $t=1,2$ and advances the understanding of Ramsey numbers for join-graphs in the asymptotic regime.

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).

Ramsey numbers of K_s + mK_t versus K_n

TL;DR

The paper resolves the growth rate of the Ramsey number for fixed by establishing the upper bound . The authors prove this via induction on , leveraging two key tools: the AKS triangle-counting lemma to convert sparse triangle structure into large independent sets, and the Li–Rouss bound to translate degree constraints into independence numbers. The result tightens previous work by eliminating a factor in the regime and achieves tightness up to constants for , aligning with Kim’s classical bound . The approach generalizes known bounds for and advances the understanding of Ramsey numbers for join-graphs in the asymptotic regime.

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).
Paper Structure (8 sections, 4 theorems, 22 equations)

This paper contains 8 sections, 4 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Useful lemmas
  3. Proof of Theorem \ref{['thm:main']}
  4. The case $s=0$
  5. The case $s=1$
  6. Bounding the maximum degree.
  7. Bounding the number of triangles.
  8. Induction step: $s\ge 2$

Key Result

Theorem 1.1

Let $m\ge 1$, $t\ge 1$ and $s\ge 0$ be fixed integers. Then the following holds. (i) $R(K_s+mK_1,K_n)\le(m+o(1))\frac{n^{s}}{(\log n)^{s-1}}.$ (ii) $R(K_s+mK_2,K_n)\le(1+o(1))\frac{n^{s+1}}{(\log n)^{s}}.$ (iii) If $t\ge3$, then $R(K_s+mK_t,K_n)=O\!\left(\frac{n^{s+t-1}\log\log n}{(\log n)^{s+t-2}}\

Theorems & Definitions (5)

  • Theorem 1.1: Liu and Li Liu-Li
  • Theorem 1.2
  • Lemma 2.1: Ajtai, Komlós and Szemerédi AKS81
  • Lemma 2.2: Li and Rousseau Li-R-96
  • Claim 1