Cutsets in \mathcal P(X)

2025/08/13 by Ginsburg, John, Sands, Bill
#03E04 #03E10 #06A06 #06D05 #06E05 #Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2508.10221

Abstract

For any set X, \mathcal P(X) denotes the collection of all subsets of X, ordered by inclusion. A \it cutset in \mathcal P(X) is a subset of \mathcal P(X) which meets every maximal chain of \mathcal P(X). A cutset is non-trivial if it does not contain X or the empty set. Our main result is the following. Theorem 1: Let X be an infinite set of cardinality κ. Every non-trivial cutset in \mathcal P(X) contains a chain of cardinality κ⁺ and an antichain of cardinality 2^κ.

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