Turán problems for hypergraphs: avoiding path or star forests

This thesis contains a number of new contributions regarding Turán problems for hypergraphs. These new results involve the Turán number of Berge linear forests, Turán numbers of star forests (in three different variants), linear Turán numbers of generalized crowns, and connected Turán numbers for Berge paths.

A classic result due to Erdos and Gallai about paths states that the number of edges e(G) of an n-vertex graph G containing no copy of a path Pk+1 of length k as its subgraph satisfies: e(G) ≤ (k−1)n/2 , with equality holding if and only if k divides n and G consists of vertex-disjoint copies of the complete graph K_k on k vertices.

It is natural to consider determining the Turán number of forests, since a forest consists of one or more trees. Motivated by the previous results, in this thesis we focus on the Turán numbers of linear forests and star forests in hypergraphs. In Chapter 2, by considering avoiding Berge linear forests in uniform hypergraphs, we determine the value of ex_r (n, Berge-L_{n,k}) for the cases 3 ≤ r ≤(k+1)/2−3 and r ≥ k+1, when k is odd, and for the cases 3 ≤ r ≤ k/2−1−\sqrt{(k+2)/2} and r ≥ k+1, when k is even, and we characterize the extremal hypergraphs. We establish an upper bound on ex_r (n, Berge-L_{n,k}) for several other cases.
In Chapter 3, we determine the Turán numbers for analogues of star forests in the hypergraph setting. In Chapter 4, we generalize the notion of a crown to r-uniform hypergraph and obtain upper and lower bounds on the Turán number for it. In Chapter 5, we consider the Turán numbers of Berge paths in connected hypergraphs and determine them asymptotically.