DEGENERATE TURAN DENSITIES OF SPARSE HYPERGRAPHS II: A SOLUTION TO THE BROWN-ERDOS-SOS PROBLEM FOR EVERY UNIFORMITY

For fixed integers r ≥ 3, e ≥ 3, and v ≥ r + 1, let fr(n, v, e) denote the maximum number of edges in an n-vertex r-uniform hypergraph in which the union of arbitrary e distinct edges contains at least v +1 vertices. In 1973, Brown, Erdos, and Sos proved that fr(n, er (e 1)k, e) = Θ (nk) and conjectured that the limit limn→ ∞ f3(n,e+2,e) n2 always exists for all fixed integers e ≥ 3. In 2020, Shangguan and Tamo conjectured that the limit limn→ ∞ fr(n,er (e 1)k,e) nk always exists for all fixed integers r > k ≥ 2 and e ≥ 3, which contains the Brown-Erdos-Sos (BES) conjecture as a special case for r = 3, k = 2. Recently, based on a result of Glock, Joos, Kim, Kühn, Lichev, and Pikhurko, Delcourt and Postle proved the BES conjecture. Extending their result, we show that the limit limn→ ∞ fr(n,er 2(e 1),e) n2 always exists, thereby resolving the BES problem for every uniformity.