2019/11/05 by Liu, Xizhi, Mubayi, Dhruv
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1911.02090
Let F be a family of r-uniform hypergraphs. The feasible region Ω(F) of F is the set of points (x,y) in the unit square such that there exists a sequence of F-free r-uniform hypergraphs whose edge density approaches x and whose shadow density approaches y. The feasible region provides a lot of combinatorial information, for example, the supremum of y over all (x,y) ∈ Ω(F) is the Turán density π(F), and Ω(∅) gives the Kruskal-Katona theorem. We undertake a systematic study of Ω(F), and prove that Ω(F) is completely determined by a left-continuous almost everywhere differentiable function; and moreover, there exists an F for which this function is not continuous. We also extend some old related theorems. For example, we generalize a result of Fisher and Ryan to hypergraphs and extend a classical result of Bollobás by almost completely determining the feasible region for cancellative triple systems.