Extremal problems for hypergraph blowups of trees

In this paper we present a novel approach in extremal set theory which may be viewed as an asymmetric version of Katona's permutation method. We use it to find more Tur\'an numbers of hypergraphs in the Erd\H{o}s--Ko--Rado range. An $(a,b)$-path $P$ of length $2k-1$ consists of $2k-1$ sets of size $r=a+b$ as follows. Take $k$ pairwise disjoint $a$-element sets $A_0, A_2, \dots, A_{2k-2}$ and other $k$ pairwise disjoint $b$-element sets $B_1, B_3, \dots, B_{2k-1}$ and order them linearly as $A_0, B_1, A_2, B_3, A_4\dots$. Define the (hyper)edges of $P_{2k-1}(a,b)$ as the sets of the form $A_i\cup B_{i+1}$ and $B_j\cup A_{j+1}$. The members of $P$ can be represented as $r$-element intervals of the $ak+bk$ element underlying set. Our main result is about hypergraphs that are blowups of trees, and implies that for fixed $k,a,b$, as $n\to \infty$ $${\rm ex}_r(n,P_{2k-1}(a,b)) = (k - 1){n \choose r - 1} + o(n^{r - 1}).$$ This generalizes the Erd\H{o}s--Gallai theorem for graphs which is the case of $a=b=1$. We also determine the asymptotics when $a+b$ is even; the remaining cases are still open.