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) [Formula presented] Extending a recent result of Glock for the special case of r=3,k=2,e=3, we show that [Formula presented] for arbitrary fixed r≥4. For the more general cases r>k≥3, we manage to show [Formula presented] 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.
- Algebraic construction
- Approximate decomposition of the complete graph
- Degenerate Turán density
- Sparse hypergraph