On the Number of Edges in a 3-Uniform Hypergraph with No (k+1)-Connected Hypersubgraphs.

Let H = (V (H) , E (H)) be a hypergraph, where V (H) is a set of vertices and E (H) is a set of non-empty subsets of V (H) called edges. If all edges of H have the same cardinality r , then H is an r -uniform hypergraph. A hypergraph H ′ = (V (H ′) , E (H ′)) is called a hypersubgraph of a hypergraph H = (V (H) , E (H)) if V (H ′) ⊆ V (H) and E (H ′) ⊆ E (H). The v e r t e x - c o n n e c t i v i t y of hypergraph H , denoted by κ (H) , is the cardinality of a minimum vertex set S such that H − S is disconnected or is a trival hypergraph. We call H k - c o n n e c t e d if κ (H) ≥ k. Tian, Lai and Meng [Y. Z. Tian, H.-J. Lai, J. X. Meng, On the sizes of vertex- k -maximal r -uniform hypergraphs, Graphs and Combinatorics 35(5) (2019) 1001–1010] conjectered that, for sufficiently large n , every n -vertex r -uniform hypergraph with no (k + 1) -connected hypersubgraphs has at most n r − n − k r + n k − 2 k r edges. This upper bound is equal to 1 2 k (n − k) 2 + 2 3 k 2 (n − k) − 3 2 k − 2 9 (n − k) when r = 3. In this paper, we prove that for k ≥ 3 and n ≥ 2. 6 7 k , every n -vertex 3 -uniform hypergraph with no (k + 1) -connected hypersubgraphs has at most 3 5 k (n − k) 2 + 1 2 k 2 (n − k) − k (n − k) + 1 3 (n − k) edges. [ABSTRACT FROM AUTHOR]