Ramsey numbers of K_s + mK_t versus K_n

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).

💡 Research Summary

The paper investigates the Ramsey number R(G, H) where G is the join of a clique K_s and m vertex‑disjoint copies of a clique K_t, denoted K_s + mK_t, and H is the complete graph K_n. For fixed integers m ≥ 1, t ≥ 1 and s ≥ 0, the authors prove the upper bound

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