On the spectrum and structure of blowup thresholds

Abstract

The chromatic threshold of Erd\H{o}s and Simonovits asks when a minimum-degree condition forces every \(H\)-free graph to have bounded chromatic number. Thomassen's homomorphism threshold strengthens this by requiring a bounded \(H\)-free homomorphic image. The recently introduced blowup threshold \(\delta_{\mathrm B}(H)\) asks for a still more rigid conclusion: when must every sufficiently dense maximal \(H\)-free graph be an actual blowup of a bounded graph? Thus the blowup threshold measures when quotient-level structure can be upgraded to exact bounded-template structure. We show that, although chromatic and homomorphism thresholds are often hard to separate, the stronger blowup threshold diverges from the chromatic threshold in several fundamental ways. First, we prove that \(\delta_{\mathrm B}(H)>0\) for every non-bipartite graph \(H\). Hence, unlike the chromatic threshold, the blowup threshold never vanishes outside the bipartite world. Second, we prove that $\delta_{\mathrm B}$ is not monotone under taking induced subgraphs. This shows that the blowup threshold is sensitive to global features of the forbidden graph and cannot be classified by a direct analogue of the monotonicity-based strategy used for chromatic thresholds. Third, we prove that $\delta_{\mathrm B}(H)=\frac{1}{4}$ for a natural family of \(3\)-chromatic constrained blowups of odd cycles. This gives a new exact blowup-threshold value beyond the chromatic-threshold spectrum.