Alexandr Kostochka (UIUC)
Extremal problems for (a,b)-paths in (a+b)-uniform hypergraphs
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Abstract: An (a,b)-path of length k is an (a+b)-uniform hypergraph P_k(a,b) defined as follows: V(P_k(a,b)) consists of k+1 sets A_1,...,A_{k+1} with |A_1|=|A_3|=...=a and |A_2|=|A_4|=...=b, and E(P_k(a,b))={A_i\cup A_{i+1}: 1 \leq i \leq k}.
Our main result is about hypergraphs that are blowups of trees, and implies that for fixed k, a, and b, if k is odd or a>b, then as n approaches infinity, ex_r(n,P_{2k-1}(a,b)) = (k - 1){n \choose r - 1} + o(n^{r - 1}). Many cases are still open.
This is joint work with Z. Füredi, T. Jiang, D. Mubayi, and J. Verstraëte.
