Kohayakawa–Prömel–Rödl conjecture for induced Ramsey numbers

Let GG be a fixed graph, and let HH be a graph on nn vertices. The quantity rind(G,H)r_{\mathrm{ind}}(G,H) is the minimum order of a graph whose every red-blue edge-coloring contains either a red induced copy of HH or a blue induced copy of GG. Kohayakawa–Prömel–Rödl conjecture. There is a constant f=f(G)f=f(G) depending only on GG such that

rind(G,H)≤nf.r_{\mathrm{ind}}(G,H)\leq n^f.

This conjecture predicts a polynomial upper bound when one graph is fixed and the other varies with order nn. The supplied text gives no resolution status.

References

Primary source

Chuang Zhong, Masaki Kashima, Yaping Mao and Yan Zhao, “Induced Ramsey numbers for fans”, arXiv:2603.19638 (2026).

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