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

@article{Bloom2018RamseyEO, title={Ramsey equivalence of \$K\_n\$ and \$K\_n+K\_\{n-1\}\$}, author={Thomas F. Bloom and Anita Liebenau}, journal={The Electronic Journal of Combinatorics}, year={2018} }

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…

## 3 Citations

Chromatic number is Ramsey distinguishing

- Computer Science, MathematicsJ. Graph Theory
- 2022

This paper shows that the chromatic number is a Ramsey distinguishing parameter and extends this to the multi-colour case and uses a similar idea to find another graph parameter which is Ramsey distinguishing.

On minimal Ramsey graphs and Ramsey equivalence in multiple colours

- Mathematics, Computer ScienceCombinatorics, Probability and Computing
- 2020

It is shown that two graphs H 1 and H 2 are q-Equivalent for even q if they are 2-equivalent, and that in general q-equivalence for some q ⩾ 3 does not necessarily imply 2-Equivalence.

Minimal Ordered Ramsey Graphs

- Mathematics, Computer ScienceDiscret. Math.
- 2020

It is proved that each Ramsey finite (not necessarily connected) ordered graph H has a pseudoforest as a Ramsey graph and therefore is a star forest with strong restrictions on the positions of the centers of the stars.

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