Linear Turán Bounds for Hypertrees and Tight Bounds for 2P2 in Linear 3-Graph

Let H be an m-uniform hypergraph (m-graph), where m≥2. If any two hyperedges in H share at most one vertex, then H is an m-uniform linear hypergraph. Let F be a given family of m-graphs. H is called F-free if H does not contain any hypergraph in F as it is subhypergraph. The linear Turán number exmlin(n,F) of F is defined as the maximum number of hyperedges of all F-free m-graphs on n vertices. Let Tk denote the m-uniform hypertree with k hyperedges. This paper studies the linear Turá number exmlin(n,F) in linear m-uniform hypergraphs. For any hypertree Tk, we have exmlin(n,Tk)≤((m−1)k−2m+3)n, which generalizes the existing results on linear triple systems to general m-uniform hypergraphs. In addition, for the m-uniform hyperpath P4 with four hyperedges, we obtain exmlin(n,P4)≤2n. It is worth mentioning that, by developing a new method, we investigate the linear Turán number of 2P2 in linear triple systems and present a tight upper bound.