Determining the Largest Intersecting Family of k-Element Subsets with Covering Number 3

Core Concepts

The author determines the size and structure of the largest intersecting family of k-element subsets of [n] with covering number 3, for any k ≥ 100 and n > 2k.

Abstract

The key insights and highlights of the content are: The author studies intersecting families of k-element subsets of [n] with covering number 3. The covering number of a family is the size of the smallest set that intersects all sets in the family. The author proves that for any k ≥ 100 and n > 2k, the largest intersecting family with covering number 3 is the family C3(n, k), which is defined in the content. The proof is divided into two cases based on whether the diversity of the family (a measure of how far the family is from a star) is large or small. For large diversity, the author shows that the size of the family is much smaller than C3(n, k) by using a "peeling" procedure to control the structure of the family. For small diversity, the author uses a "bipartite switching" technique to transform the family into one that is isomorphic to the optimal family T2(k), while preserving the covering number. The author also discusses the challenges in extending these results to families with covering number greater than 3, which remains an open problem.

Stats

The following key metrics and figures are used in the content: The size of the largest intersecting family C3(n, k) is given by the expression in (2.7). The author provides a lower bound on |C3(n, k)| in (2.11) and (2.12). The author bounds the size of the family F using the peeling procedure, obtaining the bound (4k + 250) * (n-3)/(k-3).

Quotes

"Frankl proved this theorem using the Delta-system method, which was behind many of the breakthroughs in extremal set theory in the 1970s and 80s." "The main difficulty for t ≥ 5 lies in the following problem: Given an intersecting family F of k-sets with τ(F) = t, what is the maximum number of hitting sets of size t it may have?"

Key Insights Distilled From

Intersecting families with covering number $3$

by Andrey Kupav... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.02621.pdf

Deeper Inquiries

What are some potential applications of the results on intersecting families with covering number 3 in other areas of computer science or mathematics

The results on intersecting families with covering number 3 have potential applications in various areas of computer science and mathematics. One application could be in coding theory, specifically in the design of error-correcting codes. By understanding the structure and properties of intersecting families with covering number 3, researchers can potentially develop more efficient coding schemes that can correct errors in data transmission or storage. Another application could be in algorithm design, particularly in the development of algorithms for set cover problems. The insights gained from studying intersecting families with covering number 3 can be utilized to optimize algorithms that involve covering a set with the minimum number of subsets, which is a common problem in computational complexity theory.

How might the techniques developed in this paper, such as the peeling procedure and bipartite switching, be extended to study intersecting families with higher covering numbers

The techniques developed in this paper, such as the peeling procedure and bipartite switching, can be extended to study intersecting families with higher covering numbers by adapting the methods to handle more complex structures. For intersecting families with covering number 4 or higher, the peeling procedure can be modified to handle additional layers of sets and coverings. Bipartite switching can also be generalized to deal with more intricate relationships between sets in the families. By enhancing and refining these techniques, researchers can explore a wider range of intersecting families with different covering numbers, providing deeper insights into the combinatorial properties of these structures.

Are there any connections between the problems studied in this paper and other areas of combinatorics, such as extremal graph theory or Ramsey theory

There are connections between the problems studied in this paper and other areas of combinatorics, such as extremal graph theory and Ramsey theory. In extremal graph theory, the concept of intersecting families can be related to the study of graph properties and structures that satisfy certain intersection properties. Understanding the extremal properties of intersecting families with specific covering numbers can provide insights into extremal graph structures with similar properties. In Ramsey theory, which deals with the emergence of order in large structures, the results on intersecting families with covering number 3 can be linked to the study of Ramsey numbers and the existence of specific combinatorial structures within larger systems. By establishing connections between these areas, researchers can deepen their understanding of the underlying principles governing combinatorial structures and their properties.