1. SATURATION FOR THE BUTTERFLY POSET
- Author
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Maria-Romina Ivan, Ivan, Maria [0000-0003-0817-3777], and Apollo - University of Cambridge Repository
- Subjects
General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Upper and lower bounds ,Combinatorics ,05D05 ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Saturation (chemistry) ,Partially ordered set ,Mathematics - Abstract
Given a finite poset $\mathcal P$, we call a family $\mathcal F$ of subsets of $[n]$ $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The induced saturated number of $\mathcal P$, denoted by $\text{sat}^*(n,\mathcal P)$, is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. In this paper we are mainly interested in the four-point poset called the butterfly. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan showed that the saturation number for the butterfly lies between $\log_2{n}$ and $n^2$. We give a linear lower bound of $n+1$. We also prove some other results about the butterfly and the poset $\mathcal N$., Comment: 13 pages
- Published
- 2023
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