Multicolour induced size Ramsey conjecture for bounded-degree trees

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Let cDelta\textgreater=1cDelta\textgreater=1, and let TT be a tree on nn vertices with maximum degree cDelta(T)\textgreater=cDeltacDelta(T)\textgreater= cDelta. For q\textgreater=1q\textgreater=1, write crhatindq(T)crhat_{\mathrm{ind}}^q(T) for the qq-colour induced size Ramsey number and crhatq(T)crhat^q(T) for the qq-colour size Ramsey number. Multicolour induced size Ramsey conjecture.

r^indq(T)=OΔ(r^q(T)).\hat r_{\mathrm{ind}}^q(T)=O_\Delta(\hat r^q(T)).

The conjecture asks whether, for bounded-degree trees, induced and ordinary size Ramsey numbers have the same order as functions of the number of colours. The source gives no resolution.

References

Primary source

António Girão and Eoin Hurley, “Embedding induced trees in sparse expanding graphs”, arXiv:2406.04260 (2024).

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