Determine the exponent of the Erdős–Rogers function f_{3,5}(n)

Determine the asymptotic exponent of the Erdős–Rogers function f_{3,5}(n), whose current bounds are n^{5/12+o(1)}\le f_{3,5}(n)\le n^{3/7+o(1)}.

Background

The Erdős–Rogers function f_{\ell,s}(n) is the largest order of a K_{\ell}-free induced subgraph guaranteed in every K_s-free graph on n vertices. The paper identifies the parameter pair (\ell,s)=(3,5) as the smallest case in which the exponent governing the asymptotic growth of f_{\ell,s}(n) is unresolved.

For this case, the paper records Sudakov’s lower bound f_{3,5}(n)\ge n{5/12+o(1)} and the best known upper bound f_{3,5}(n)\le n{3/7+o(1)}, leaving a gap between the two exponents. The paper’s theorem matches the existing upper bound for this pair but does not determine the exact exponent.

References

It is perhaps interesting to discuss the somewhat notorious $(\ell,s)=(3,5)$ case, which is the smallest case of the Erdős--Rogers problem for which the exponent is unknown. Sudakov's long-standing lower bound gives $f_{3,5}(n)\geq n{5/12+o(1)}$, while the current best upper bound, due to Janzer and Sudakov, gives $f_{3,5}(n)\leq n{3/7+o(1)}$.

— On the asymptotics of the Erdős-Rogers function  (2609.37987 - Bradač et al., 29 Sep 2026) in Section 1, Introduction