Exact solution of the hypergraph Turán problem for k-uniform linear paths

Abstract

A k-uniform linear path of length ℓ, denoted by ℙ ^(k)_ℓ , is a family of k-sets {F ₁,...,F _ℓ such that |F _i ∩ F _i+1|=1 for each i and F _i ∩ F _bj = $\not 0$ whenever |i−j|>1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turán number of H, denoted by ex _k (n, H), is the maximum number of edges in a k-uniform hypergraph $\mathcal{F}$ on n vertices that does not contain H as a subhypergraph. With an intensive use of the delta-system method, we determine ex _k (n, P ^(k)_ℓ exactly for all fixed ℓ ≥1, k≥4, and sufficiently large n. We show that

$ex_k (n,\mathbb{P}_{2t + 1}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} )$

.

The only extremal family consists of all the k-sets in [n] that meet some fixed set of t vertices. We also show that

$ex(n,\mathbb{P}_{2t + 2}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} ) + (_{k - 2}^{n - t - 2} )$

, and describe the unique extremal family. Stability results on these bounds and some related results are also established.