Homomorphism and VC-dimension thresholds: spectra and separations

Abstract

Minimum-degree thresholds ask when excluding a fixed graph $H$ forces a dense graph to admit a simple global description. For each fixed chromatic number, the chromatic threshold has only three possible values. We show that this finite-spectrum phenomenon is special to chromatic threshold: already among $3$-chromatic graphs, both the homomorphism and VC-dimension thresholds have infinite spectra and are nonmonotone under taking induced subgraphs. For complete tripartite graphs with a singleton part, we prove ${\delta }_{\mathrm{hom}}(K_{1,s,t})\ge \max \left\{\frac{1}{3},\frac{s}{1+s+t}\right\}$, with equality for an infinite range of $s,t$; in particular, ${\delta }_{\mathrm{hom}}(K_{1,s,s})=s/(2s+1)$ for every $s\ge 2$. More generally, for every $r\ge 3$, the value $(r-2)/(r-1)$ is an accumulation point of the homomorphism thresholds of $r$-chromatic graphs. For maximal $H$-free graphs, we determine the VC-dimension threshold of every complete tripartite graph and prove that it is positive for every nonbipartite $H$, yielding in particular the exact value for every odd cycle. We also classify the chromatic threshold under an a priori VC-dimension bound. Together with known blowup-threshold results, our theorems reveal that ${\delta }_{\chi },{\delta }_{\mathrm{hom}},{\delta }_{\mathrm{VC}}$, and ${\delta }_{\mathrm{B}}$ are \emph{pairwise distinct}: bounded colorability, homomorphic compressibility, neighborhood complexity, and exact blowup structure are genuinely different forms of global simplicity. The proofs develop random and grid-based obstructions to bounded homomorphic images, saturated gadgets that preserve high VC-dimension under maximal completion, and a core-orientation method for raising minimum degree while preserving $H$-freeness.