Alexandr Kostochka (UIUC), Extremal problems for bushes
Abstract: For t > 1, a graph bush B_{t,h} is the radius 2 tree obtained from the star K_{1,t} by joining each degree-one vertex to h new vertices. For positive integers r,a,b,t,h with r = a + b and t > 1, the (a,b,t,h)-bush B_{t,h}(a,b) is an r-uniform hypergraph obtained from the graph bush B_{t,h} by blowing up its center and leaves into a-sets and all the neighbors of the center into b-sets. We are interested in the Turán number of B_{t,h}(a,b), that is, the maximum number of edges in an n-vertex r-uniform hypergraph with no B_{t,h}(a,b) subgraph.
Since B_{t,h}(a,b) contains t disjoint edges, the Turán number of B_{t,h}(a,b) is at least (t-1)*((n-1) choose (r-1)). A recent theorem by Füredi-Jiang-Kostochka-Mubayi-Verstraëte implies that this is asymptotically sharp for a > b > 1 and for (a,b,t,h) = (1,2,2,1). The goal of this talk is to show that the Turán number of B_{t,h}(a,b) is asymptotically equal to (t-1)*((n-1) choose (r-1)) for all fixed a,b,t,h with t > 1 and a > 1. The claim also holds for (a,b,t,h) = (1,r-1,t,1) when t > 1 and r > 2.
This talk is joint work with Zoltán Füredi.
