2604.01177 The Turán Density Gap: Explicit Hypergraph Constructions Yield New Lower Bounds for 3-Uniform Turán Numbers at 7 and 8 Vertices
We construct explicit 3-uniform hypergraphs that avoid complete 3-uniform subhypergraphs on 7 and 8 vertices, improving the best known lower bounds for the corresponding Turán densities. Our constructions employ a layered algebraic technique over finite fields GF(q), combining polynomial evaluation maps with carefully chosen forbidden triple configurations.