Hypergraph Turán Problems and the Szemerédi Regularity Lemma for Sparse Graphs

The determination of Turán numbers for hypergraphs remains one of the most challenging frontiers in extremal combinatorics. While the classical Szemerédi Regularity Lemma has provided a powerful framework for analyzing dense structures, its direct applicability diminishes in the sparse regime, where edge densities vanish asymptotically. This monograph explores the intersection of Hypergraph Turán Theory and the Sparse Regularity Method, specifically focusing on the Kohayakawa-Rödl formalism and its hypergraph extensions. We investigate the transference of the supersaturation phenomenon to sparse pseudo-random hypergraphs and derive bounds for the extremal number ex(n, H) within this setting. By establishing a sparse counting lemma for linear hypergraphs, we demonstrate that the asymptotic behavior of Turán densities in sufficiently quasi-random sparse environments mirrors that of the dense case. These results elucidate the structural stability of extremal configurations under sparse random perturbations.