Finite-Kernel Extremizers in Sparse Extremal Graph Counting

Abstract

We develop a finite-kernel framework for sparse extremal graph counting. The problems considered here ask for the maximum number of copies or homomorphisms of a fixed graph under sparse edge constraints. In this regime, the leading term need not be governed by a single dense block. Instead, the extremal mass may be supported on several interacting asymptotic scales. Our framework identifies these scales via a finite-dimensional linear program, separates the leading contributions through a finite state decomposition, and synchronizes or realizes them inside a finite kernel. We apply this framework in three settings. First, we prove the sparse threshold conjecture of Day and Sarkar for graphons. For every fixed graph $H$ without isolated vertices, we prove that \[ \sup_{t(K_2,W)\le \beta} t(H,W)=\beta^{|V(H)|-\alpha^*(H)}(C_T(H)+o(1)) \] as $\beta\to0$, where $\alpha^*(H)$ is the fractional independence number of $H$ and $C_T(H)$ is an explicit sharp constant attained by a three-step threshold graphon. Second, we affirmatively answer a question of Blekherman and Patel by showing that, for every graph $H$, whenever $m\to\infty$ and $m=o(n^{3/2})$, threshold graphs asymptotically maximize $\hom(H,G)$ among all graphs with at most $n$ vertices and at most $m$ edges. Third, Gerbner, Nagy, Patk\'os, and Vizer conjectured that, among all bipartite graphs with $n$ vertices and $m$ edges, the quasi-complete bipartite graph asymptotically maximizes the number of copies of every fixed bipartite graph $H$ whenever $m=\omega(n)$ and $m\le n^2/4$. We disprove this conjecture in the subquadratic range and give the correct order of magnitude in terms of $\kappa_H(n,m)$, a finite-kernel scale defined by a finite-dimensional variational problem.