Turan numbers of r-graphs on r+1 vertices

Let H-k (R) denote an r-uniform hypergraph with k edges and r + 1 vertices, where k <= r + 1 (it is easy to see that such a hypergraph is unique up to isomorphism). The known general bounds on its Turan density are pi(H-k (R)) <= k-2/r for all k >= 3, and pi(H-3 (R)) >= 2(1-r) for k = 3. We prove that pi(H-k (R)) >= (C-k - o (1)) r(-(1+1/k-2)) as r -> infinity. In the case k = 3, we prove pi(H-3 (R)) >= (1.7215 - o(1)) r(-2) as r -> infinity, and pi(H-3 (R)) > r(-2) for all r. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.