The conjecture for even-clique Ramsey–Turán densities

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Let R(3,s)R(3,s) be the smallest integer NN such that every red-blue colouring of KNK_N contains a red K3K_3 or a blue KsK_s. Let ϱ(K3,K2s)\varrho(K_3,K_{2s}) denote the multicolour Ramsey–Turán density for triangle versus the even clique K2sK_{2s}. Even-clique Ramsey–Turán conjecture. For all s≥2s\geq2,

ϱ(K3,K2s)=12(1−1R(3,s)).\varrho(K_3,K_{2s})=\frac{1}{2}\left(1-\frac{1}{R(3,s)}\right).

This is proposed as the even-clique analogue of the previously recalled odd-clique conjecture. The provided text gives no resolution, so the conjecture remains open here.

References

Primary source

Jaehoon Kim, Younjin Kim and Hong Liu, “Two conjectures in Ramsey-Turán theory”, arXiv:1803.04721 (2018).

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