Maximum Size Intersecting Families of Bounded Minimum Positive Co-degree

Balogh, Jozsef; Lemons, Nathan Wishard; Palmer, Cory

doi:10.1137/20m1336989

Maximum Size Intersecting Families of Bounded Minimum Positive Co-degree

Journal Article · Thu Jul 01 00:00:00 EDT 2021 · SIAM Journal on Discrete Mathematics (SIDMA)

DOI:https://doi.org/10.1137/20m1336989· OSTI ID:1843168

Balogh, Jozsef ^[1]; ^[2]; ^[3]

Univ. of Illinois at Urbana-Champaign, IL (United States)
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Univ. of Montana, Missoula, MT (United States)

Let $$\mathcal{H}$$ be an $$r$$-uniform hypergraph. The minimum positive co-degree of $$\mathcal{H}$$, denoted by $$\delta_{r-1}^+(\mathcal{H})$$, is the minimum $$k$$ such that if $$S$$ is an $(r-1)$-set contained in a hyperedge of $$\mathcal{H}$$, then $$S$$ is contained in at least $$k$$ hyperedges of $$\mathcal{H}$$. For $$r\geq k$$ fixed and $$n$$ sufficiently large, we determine the maximum possible size of an intersecting $$r$$-uniform $$n$$-vertex hypergraph with minimum positive co-degree $$\delta_{r-1}^+(\mathcal{H}) \geq k$$ and characterize the unique hypergraph attaining this maximum. This generalizes the Erd\Hos--Ko--Rado theorem which corresponds to the case $k=1$. Furthermore, our proof is based on the delta-system method.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 1843168

Report Number(s):: LA-UR-20-23462

Journal Information:: SIAM Journal on Discrete Mathematics (SIDMA), Journal Name: SIAM Journal on Discrete Mathematics (SIDMA) Journal Issue: 3 Vol. 35; ISSN 0895-4801

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

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97 MATHEMATICS AND COMPUTING
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Maximum Size Intersecting Families of Bounded Minimum Positive Co-degree

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