Linear chromatic threshold for dominating complete-graph models

Determine whether there exists an absolute constant C greater than zero such that every graph with chromatic number at least Ct contains a dominating K_t-model.

Background

The paper disproves the dominating Hadwiger conjecture by constructing, for arbitrarily large t, graphs with chromatic number at least (1+ε)t that contain no dominating K_t-model. This leaves unresolved whether a weaker linear statement might hold with a sufficiently large absolute multiplicative constant C: namely, whether chromatic number at least Ct always forces a dominating K_t-model.

References

It does leave open the possibility that, for some absolute constant $C > 0$, every graph with chromatic number at least $Ct$ contains a dominating $K_t$-model.

— Disproof of the dominating Hadwiger conjecture  (2609.35361 - Illingworth et al., 28 Sep 2026) in Section 1, immediately after Theorem 1.1