2025/07/07 by Ivan, Maria-Romina, Jaffe, Sean · 3 citations
#05D05 #06A07 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.05122
For a fixed poset \mathcal P we say that a family \mathcal F⊆\mathcal P([n]) is \mathcal P-saturated if it does not contain an induced copy of \mathcal P, but whenever we add a new set to \mathcal F, we form an induced copy of \mathcal P. The size of the smallest such family is denoted by sat^*(n, \mathcal P).\par For the diamond poset \mathcal D2 (the two-dimensional Boolean lattice), while it is easy to see that the saturation number is at most n+1, the best known lower bound has stayed at O(√ n) since the introduction of the area of poset saturation. In this paper we prove that sat^*(n, \mathcal D2)≥ (n+1)/(5), establishing that the saturation number for the diamond is linear. The proof uses a result about certain pairs of set systems which may be of independent interest.