Ramsey equivalence of $K_n$ and $K_n+K_{n-1}$

  • Thomas F. Bloom
  • Anita Liebenau
Keywords: Graph theory, Ramsey theory


We prove that, for $n\geqslant 4$, the graphs $K_n$ and $K_n+K_{n-1}$ are Ramsey equivalent. That is, if $G$ is such that any red-blue colouring of its edges creates a monochromatic $K_n$ then it must also possess a monochromatic $K_n+K_{n-1}$. This resolves a conjecture of Szabó, Zumstein, and Zürcher.

The result is tight in two directions. Firstly, it is known that $K_n$ is not Ramsey equivalent to $K_n+2K_{n-1}$. Secondly, $K_3$ is not Ramsey equivalent to $K_3+K_{2}$. We prove that any graph which witnesses this non-equivalence must contain $K_6$ as a subgraph.

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