[Paper Review] Counting substructures III: quadruple systems

Dhruv Mubayi|ArXiv.org|May 28, 2009

Limits and Structures in Graph Theory15 references3 citations

TL;DR

This paper establishes asymptotically sharp lower bounds on the number of copies of specific 4-uniform hypergraphs—namely, the 2-, 3-, and 4-books and the expanded triangle—within 4-graphs that exceed the extremal Turán number by a small number of edges. Leveraging the hypergraph removal lemma and stability results for Turán problems, it proves that such hypergraphs contain at least Θ(n³) copies of the forbidden substructure, refining earlier results that only guaranteed the existence of one copy.

ABSTRACT

For various quadruple systems F, we give asymptotically sharp lower bounds on the number of copies of F in a quadruple system with a prescribed number of vertices and edges. Our results extend those of Furedi, Keevash, Pikhurko, Simonovits and Sudakov who proved under the same conditions that there is one copy of $F$. Our proofs use the hypergraph removal Lemma and stability results for the corresponding Turan problem proved by the above authors.

Motivation & Objective

• To determine asymptotically sharp lower bounds on the number of copies of specific 4-uniform hypergraphs (books P₂, P₃, P₄ and expanded triangle C₃) in hypergraphs that exceed the extremal Turán number by a small number of edges.
• To extend prior results that only guaranteed the existence of at least one copy of a forbidden substructure when exceeding the Turán number.
• To apply the hypergraph removal lemma and stability theorems to derive quantitative lower bounds on substructure counts in extremal hypergraph theory.
• To characterize the minimum number of copies of each forbidden substructure that must appear when adding a small number of edges to the extremal hypergraph.

Proposed method

• Utilizes the hypergraph removal lemma, which states that if a hypergraph contains few copies of a forbidden substructure, then a small number of edge deletions can destroy all copies.
• Applies stability theorems for Turán problems to identify the unique extremal hypergraphs (T⁴(n), D⁴(n), B⁴(n)) that maximize edges without containing the forbidden substructures.
• Employs double counting and degree-based arguments to estimate the number of copies of substructures like Pₗ and C₃ in hypergraphs with slightly more edges than the extremal number.
• Analyzes vertex degrees and edge distributions in the hypergraph, partitioning edges into good and bad sets to derive contradictions under assumptions of low substructure count.
• Uses bipartite graph constructions between the set of edges in the residual hypergraph and a matching to show that some pair of vertices must have high degree in the residual, leading to a contradiction if substructure count is too low.
• Employs case analysis on edge partitions and pair intersections to bound the number of copies of C₃ involving edges from the main and residual hypergraphs.

Experimental results

Research questions

• RQ1 What is the minimum number of copies of a 4-uniform hypergraph F that must exist in a 4-graph with ex(n,F) + q edges, for small q?
• RQ2 How does the hypergraph removal lemma help in transforming qualitative existence results into quantitative lower bounds on substructure counts?
• RQ3 Can stability results for extremal 4-graphs be used to derive asymptotically sharp lower bounds on the number of copies of F when the edge count slightly exceeds the Turán threshold?
• RQ4 For which 4-graphs F is the extremal hypergraph unique up to isomorphism, and how does this uniqueness enable precise counting of substructures upon edge addition?
• RQ5 What is the precise asymptotic growth rate of the number of copies of F in a hypergraph that is just above the Turán threshold?

Key findings

• For each of the 4-graphs P₂, P₃, P₄, and C₃, the number of copies in a 4-graph with ex(n,F) + q edges (for small q) is at least Θ(n³), establishing asymptotically sharp lower bounds.
• The minimum number of copies c(n,F) of F in a hypergraph formed by adding one edge to the extremal hypergraph H(n,F) is Θ(n³), with explicit lower-order terms given for P₂, P₃, and P₄.
• The proof shows that if a 4-graph has slightly more than ex(n,F) edges, it must contain Ω(n³) copies of F, contradicting the assumption of few copies via degree and counting arguments.
• The existence of a vertex with degree Ω(n³) in the residual hypergraph (after removing the main extremal structure) is shown to be necessary, leading to a contradiction if substructure count is too low.
• The argument relies on the fact that each edge in the residual hypergraph contributes many copies of C₃ when combined with edges from the main hypergraph, forcing a high edge count in the residual.
• A contradiction is derived by showing that some vertex must have degree Ω(n³) in the residual hypergraph, violating the stability assumption that the residual structure is small.