Maximum Erdős-Ko-Rado sets of chambers and their antidesigns in vector-spaces of even dimension

2024/06/02 by Heering, Philipp, Lansdown, Jesse, Metsch, Klaus · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2406.00740

Abstract

A chamber of the vector space \mathbbF_qⁿ is a set \S₁,…,S_n-1\ of subspaces of \mathbbF_qⁿ where S₁⊂ S₂⊂ \dotso ⊂ S_n-1 and dim(S_i)=i for i=1,…,n-1. By Γ_n(q) we denote the graph whose vertices are the chambers of \mathbbF_qⁿ with two chambers C₁=\S₁,…,S_n-1\ and C₂=\T₁,…,T_n-1\ adjacent in Γ_n(q), if S_i∩ T_n-i=\0\ for i=1,…,n-1. The Erdős-Ko-Rado problem on chambers is equivalent to determining the structure of independent sets of Γ_n(q). The independence number of this graph was determined in [7] for n even and given a subspace P of dimension one, the set of all chambers whose subspaces of dimension \frac n2 contain P attains the bound. The dual example of course also attains the bound. It remained open in [7] whether or not these are all maximum independent sets. Using a description from [6] of the eigenspace for the smallest eigenvalue of this graph, we prove an Erdős-Ko-Rado theorem on chambers of \mathbbF_qⁿ for sufficiently large q, giving an affirmative answer for n even.

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