Degenerate Turán densities of sparse hypergraphs

Abstract

For fixed integers r>k≥2,e≥3, let fr(n,er−(e−1)k,e) be the maximum number of edges in an r-uniform hypergraph in which the union of any e distinct edges contains at least er−(e−1)k+1 vertices. A classical result of Brown, Erdős and Sós in 1973 showed that fr(n,er−(e−1)k,e)=Θ(nk). The degenerate Turán density is defined to be the limit (if it exists)π(r,k,e):=limn→∞⁡fr(n,er−(e−1)k,e)nk. Extending a recent result of Glock for the special case of r=3,k=2,e=3, we show thatπ(r,2,3):=limn→∞⁡fr(n,3r−4,3)n2=1r2−r−1 for arbitrary fixed r≥4. For the more general cases r>k≥3, we manage to show1rk−r≤liminfn→∞fr(n,3r−2k,3)nk≤limsupn→∞fr(n,3r−2k,3)nk≤1k!(rk)−k!2, where the gap between the upper and lower bounds are small for r≫k.The main difficulties in proving these results are the constructions establishing the lower bounds. The first construction is recursive and purely combinatorial, and is based on a (carefully designed) approximate induced decomposition of the complete graph, whereas the second construction is algebraic, and is proved by a newly defined matrix property which we call strongly 3-perfect hashing.