On the Erdős-Ko-Rado problem of flags with type \1, n-3 \ of finite sets

2025/06/25 by Heering, Philipp
#05C35 #05C69 #05D05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.20556

Abstract

A flag of a finite set S is a set f of non-empty, proper subsets of S, such that X⊆ Y or Y⊆ X for all X,Y∈ f. Two flags f₁ and f₂ of S are opposite if X₁∩ X₂=∅, or X₁∪ X₂=S for all X₁∈ f₁ and X₂∈ f₂. The set \|X| | X∈ f \ is the type of a flag f. A set of pairwise non-opposite flags is an Erdős-Ko-Rado set. In 2022 Metsch posed the problem of determining the maximum size of all Erdős-Ko-Rado sets of flags of type T with |T|=2. We contribute towards this by determining the maximum size for flags of type \ 1,n-3\ for finite sets with n elements. Furthermore we answer an open questions of Metsch regarding a small case.

Citations

• The largest sets of non-opposite chambers in spherical buildings of type B
• Maximum Erdős-Ko-Rado sets of chambers and their antidesigns in vector-spaces of even dimension
• On the largest independent sets in the Kneser graph on chambers of PG(4,q)
• The minimum weight of the code of intersecting lines in \rm PG(3,q)
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• Erdös-Ko-Rado sets of flags of finite sets
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• An algebraic approach to Erdős-Ko-Rado sets of flags in spherical buildings
• On the chromatic number of two generalized Kneser graphs
• Maximal sets of k-spaces pairwise intersecting in at least a (k-2)-space
• An algebraic proof of the Erdős-Ko-Rado theorem for intersecting families of perfect matchings
• INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS

Related

• Erdös-Ko-Rado sets of flags of finite sets
• An algebraic approach to Erdős-Ko-Rado sets of flags in spherical buildings II
• Simultaneous generating sets for flags
• The Rado Multiplicity Problem in Vector Spaces over Finite Fields
• Elementary Techniques for Erdos-Ko-Rado-like Theorems