The conjectured exact value of the Ramsey parameter f(k)

From papers

Let f(k)f(k) be the Ramsey parameter defined in the source: the minimum parameter governing the relevant dependency-digraph Ramsey problem, and let g(k)g(k) be the corresponding ordered-graph parameter. The exact-value conjecture for f(k)f(k).

f(k)=k24+1.f(k)=\left\lfloor\frac{k^2}{4}\right\rfloor+1.

The source notes that this would imply the optimal equality g(k)=f(k)=k24+1g(k)=f(k)=\left\lfloor\frac{k^2}{4}\right\rfloor+1, and reports computer verification for k6k\le 6; the conjecture is otherwise unresolved.

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Sources & referencesView supporting material

Primary source

Domagoj Bradač, Hong Liu, Zhuo Wu and Zixiang Xu, “Clique density vs blowups”, arXiv:2410.07098 (2024).

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