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Improved bounds on the maximum diversity of intersecting families

EUROPEAN JOURNAL OF COMBINATORICS (2024)

Intersections and distinct intersections in cross-intersecting families

Peter Frankl et al.

Summary: This paper investigates two cross-intersecting families of k-subsets and provides the families that maximize the intersections and distinct intersections respectively. The paper also studies the problem of k-subset families under a fixed T.

EUROPEAN JOURNAL OF COMBINATORICS (2023)

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A product version of the Hilton-Milner theorem

Peter Frankl et al.

JOURNAL OF COMBINATORIAL THEORY SERIES A (2023)

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On the Maximum of the Sum of the Sizes of Non-trivial Cross-Intersecting Families

P. Frankl

Summary: This article introduces the concept of cross-intersecting and the definition of non-trivial families. It establishes the best possible bound between F and G when they are non-trivial and cross-intersecting, and presents a strengthened shifting technique for the proof. The most general result is Theorem 4.1.

COMBINATORICA (2023)

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Diversity

Peter Frankl et al.

Summary: This paper investigates the application of diversity in different contexts of extremal set theory problems, particularly in the context of cross-intersecting families and the Kruskal-Katona theorem. The main result includes a sharp stability result for cross-intersecting families in terms of diversity, and a slightly more general result for the Kruskal-Katona theorem. Additionally, the paper addresses the question of diversity in non-uniform intersecting families raised by Huang.

JOURNAL OF COMBINATORIAL THEORY SERIES A (2021)

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