Many cliques in bounded-degree hypergraphs

Abstract

Recently Chase determined the maximum possible number of cliques of size t in a graph on n vertices with given maximum degree. Soon afterward, Chakraborti and Chen answered the version of this question in which we ask that the graph have m edges and fixed maximum degree (without imposing any constraint on the number of vertices). In this paper we address these problems on hypergraphs. For s-graphs with s ≥ 3 a number of issues arise that do not appear in the graph case. For instance, for general s-graphs we can assign degrees to any i-subset of the vertex set with 1 ≤ i ≤ s − 1.We establish bounds on the number of t-cliques in an s-graph H with i-degree bounded by ∆ in three contexts: H has n vertices; H has m (hyper)edges; and (generalizing the previous case) H has a fixed number p of u-cliques for some u with s ≤ u ≤ t. When ∆ is of a special form we characterize the extremal s-graphs and prove that the bounds are tight. These extremal examples are the shadows of either Steiner systems or packings. On the way to proving our uniqueness results, we extend results of Füredi and Griggs on uniqueness in Kruskal-Katona from the shadow case to the clique case.

A localized approach to generalized Turán problems

“…We write K t (H) for the set of t-cliques in H, i.e., K t (H) = {S ⊆ V (H) : H[S] ∼ = K (q) t }. We use the following upper bound on N (K (q) t , H), which is proved in [16,Theorem 32] as an immediate consequence of Lovász' approximate version of the Kruskal-Katona theorem.…”

“…Design theory provides an infinite family of graphs that meet this bound; we direct the reader to [16] for more information on such hypergraphs. If H is a q-shadow of a Steiner system S(i, r, n) for some r then by [16,Lemma 38(b)] we have x(I) = r for every I and x(T ) = r for every T , so…”

“…It seems interesting and challenging to characterize all of the extremal q-graphs in Theorem 5.8. See [16,Theorem 43] for a related characterization of the extremal q-graphs in the non-localized theorem.…”

“…As a corollary of Theorem 5.8 we obtain the following theorem of Radcliffe and the first author on maximizing the number of t-cliques among bounded-degree q-uniform hypergraphs. Theorem 5.9 (Kirsch and Radcliffe [16]). Let 1 ≤ i < q ≤ t and suppose H is an q-uniform hypergraph on n vertices such that ∆ i (H) ≤ x−i q−i for some real number x ≥ q.…”

“…The following conjecture is a localized form of a theorem of Frohmader [10], as phrased in [16,Theorem 8], on maximizing the number of t-cliques among m-edge, K r+1 -free graphs. .…”