On Hypergraphs of Girth Five

Abstract

In this paper, we study $r$-uniform hypergraphs ${\cal H}$ without cycles of length less than five, employing the definition of a hypergraph cycle due to Berge. In particular, for $r = 3$, we show that if ${\cal H}$ has $n$ vertices and a maximum number of edges, then $$|{\cal H}|={\textstyle 1\over6}n^{3/2} + o(n^{3/2}).$$ This also asymptotically determines the generalized Turán number $T_{3}(n,8,4)$. Some results are based on our bounds for the maximum size of Sidon-type sets in $\Bbb{Z}_{n}$.

Linearity of saturation for Berge hypergraphs

“…[18,20,19,22,24]) or Berge-F (c.f. [10,12,13,11,21,26,9]) for various graphs F has received quite a lot of attention recently.…”

On the independence number of the Erdős‐Rényi and projective norm graphs and a related hypergraph

“…Such exact results are relatively rare in extremal graph theory. The graph ER q has also been used to solve a similar problem for hypergraphs (see [19]). These hypergraphs will be dealt with in the last section.…”

“…In [19], Lazebnik and Verstraëte (using an idea of Lovász, see Aknowledgment in [19]) construct a series of hypergraphs H q of girth 5. These hypergraphs are used to determine the asymptotics of the Turán number T 3 (n, 8,4), defined as the maximum number of edges in a 3-graph on n vertices in which no set of 8 vertices spans more than 4 edges.…”

“…We are to show that α(H q ) ≤ q 2 /2 + q 3/2 + O(q). To facilitate the proof we will use a few facts about the graph ER o q : the graph is q + 1 regular on q 2 + q + 1 vertices, every edge which does not have an absolute endvertex is contained in a unique triangle (see [19]), and a non-absolute vertex is connected to either 0 or 2 absolute vertices. Recall that the vertex set of H q is the set of non-absolute points in V (ER q ).…”

Nearly-Regular Hypergraphs and Saturation of Berge Stars

“…Based on this, we can define the k-uniform extremal number ex k (n, Berge-F ) and saturation number sat k (n, Berge-F ) to be the maximum, and respectfully minimum, number of edges in a Berge-F -saturated k-uniform hypergraph on n vertices. Extremal numbers for Berge hypergraphs have been studied extensively, [18,14,6,15,12,20,13]. On the other hand, saturation numbers for Berge hypergraphs have been mostly left untouched.…”