2026/01/24 by Rajat Adak, Pragya Verma · 1 citation
#math.CO #cs.DM
A hypergraph H is said to be linear if every pair of vertices lies in at most one hyperedge. Given a family F of r-uniform hypergraphs, an r-uniform hypergraph H is said to be \emphF-free if it contains no member of F as a subhypergraph. The linear Turán number exrlin(n,F) denotes the maximum number of hyperedges in an F-free linear r-uniform hypergraph on n vertices. Gyárfás, Ruszinkó, and Sárközy~[Linear Turán numbers of acyclic triple systems, European J. Combin. (2022)] initiated the study of bounds on the linear Turán number for acyclic 3-uniform linear hypergraphs. In this paper, we extend the study of linear Turán numbers for acyclic systems to higher uniformity. We first give a construction for linear r-uniform trees with k edges that yields the lower bound exrlin(n,Tkr)≥ n(k-1)/r, under mild divisibility and existence assumptions. Next, we study hypertrees with four edges. We prove the exact bound exrlin(n,B4r)≤ (r+1)n/r and characterize the extremal hypergraph class, where B4r is formed from S3r by appending a hyperedge incident to a degree-one vertex. We also prove the bound exrlin(n,E4r)≤ (2r-1)n/r for the crown E4r. Finally, we give a construction showing exrlin(n,P4r)≥ (r+1)n/r under suitable assumptions and conclude with a conjecture on sharp upper bound for P4r.