On the Boundedness of Degenerate Hypergraphs

Abstract

We investigate the impact of a high-degree vertex in Turán problems for degenerate hypergraphs (including graphs). We say an r-graph F is bounded if there exist constants α, β > 0 such that for large n, every n-vertex F-free r-graph with a vertex of degree at least \(\alpha \left({\matrix{{n - 1} \cr {r - 1}}}\right)\) has fewer than (1 − β) · ex(n, F) edges. The boundedness property is crucial for recent works that aim to extend the classical Hajnal–Szemerédi Theorem (Toward a density Corrádi–Hajnal theorem for degenerate hypergraphs. J. Combin. Theory Ser. B, 172, 221–262 (2025)) and the anti-Ramsey theorems of Erdős–Simonovits–Sós (Tight bounds for rainbow partial F-tiling in edge-colored complete hypergraphs. J. Graph Theory, 110(4), 457–467 (2025)). We show that many well-studied degenerate hypergraphs, such as all even cycles, most complete bipartite graphs, and the expansion of most complete bipartite graphs, are bounded. In addition, to prove the boundedness of the expansion of complete bipartite graphs, we introduce and solve a Zarankiewicz-type problem for 3-graphs, strengthening a theorem by Kostochka–Mubayi–Verstraëte (Turán problems and shadows III: expansions of graphs. SIAM J. Discrete Math., 29(2), 868–876 (2015)).