Bounded-degree function for clique-free derangement graphs

Determine whether there exists a function f: N -> N such that every transitive permutation group of degree n whose derangement graph lacks a clique of size k satisfies n <= f(k).

Background

The paper studies cliques in derangement graphs of finite transitive permutation groups. Earlier work established that every transitive group of degree at least 3 has a triangle in its derangement graph, yielding a bound for the case of missing triangles. Question 6.1 of [19] asks whether the absence of larger cliques similarly forces the permutation degree to be bounded as a function of the clique size.

The present paper proves a substantial partial result: apart from explicitly listed low-degree or exceptional groups, every transitive group has a clique of size at least four. Thus the question remains unresolved in general, while the paper supplies further evidence for an affirmative answer.

References

Question 6.1 of [19] asks whether a function f : N -> N exists such that if G is transitive of degree n and IG lacks a k-clique, then n ≤ f(k).

— Kronecker classes and cliques in derangement graphs  (2502.01287 - Cazzola et al., 3 Feb 2025) in Section 1, immediately before Theorem 1.1; cited as Question 6.1 of [19]