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

Abstract

Let [Formula: see text] be a hypergraph, where [Formula: see text] is a set of vertices and [Formula: see text] is a set of non-empty subsets of [Formula: see text] called edges. If all edges of [Formula: see text] have the same cardinality [Formula: see text], then [Formula: see text] is an [Formula: see text]-uniform hypergraph. A hypergraph [Formula: see text] is called a hypersubgraph of a hypergraph [Formula: see text] if [Formula: see text] and [Formula: see text]. The [Formula: see text]-[Formula: see text] of hypergraph [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum vertex set [Formula: see text] such that [Formula: see text] is disconnected or is a trival hypergraph. We call [Formula: see text] [Formula: see text]-[Formula: see text] if [Formula: see text]. Tian, Lai and Meng [Y. Z. Tian, H.-J. Lai, J. X. Meng, On the sizes of vertex-[Formula: see text]-maximal [Formula: see text]-uniform hypergraphs, Graphs and Combinatorics 35(5) (2019) 1001–1010] conjectered that, for sufficiently large [Formula: see text], every [Formula: see text]-vertex [Formula: see text]-uniform hypergraph with no [Formula: see text]-connected hypersubgraphs has at most [Formula: see text] edges. This upper bound is equal to [Formula: see text] when [Formula: see text]. In this paper, we prove that for [Formula: see text] and [Formula: see text], every [Formula: see text]-vertex [Formula: see text]-uniform hypergraph with no [Formula: see text]-connected hypersubgraphs has at most [Formula: see text] edges.