Geodesic
Improved bounds for cross-Sperner systems
Natalie Behague1 ; Akina Kuperus1 ; Natasha Morrison ; Ashna Wright1
1University of Victoria
The electronic journal of combinatorics, Tome 31 (2024) no. 2
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A collection of families $(\mathcal{F}_{1}, \mathcal{F}_{2} , \cdots , \mathcal{F}_{k}) \in \mathcal{P}([n])^k$ is cross-Sperner if there is no pair $i \not= j$ for which some $F_i \in \mathcal{F}_i$ is comparable to some $F_j \in \mathcal{F}_j$. Two natural measures of the 'size' of such a family are the sum $\sum_{i = 1}^k |\mathcal{F}_i|$ and the product $\prod_{i = 1}^k |\mathcal{F}_i|$. We prove new upper and lower bounds on both of these measures for general $n$ and $k \ge 2$ which improve considerably on the previous best bounds. In particular, we construct a rich family of counterexamples to a conjecture of Gerbner, Lemons, Palmer, Patkós, and Szécsi from 2011.
DOI : 10.37236/11860
Classification : 05D05
Mots-clés : extremal set systems, Sperner property

Natalie Behague  1   ; Akina Kuperus  1   ; Natasha Morrison    ; Ashna Wright  1

1 University of Victoria 
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     title = {Improved bounds for {cross-Sperner} systems},
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Natalie Behague; Akina Kuperus; Natasha Morrison; Ashna Wright. Improved bounds for cross-Sperner systems. The electronic journal of combinatorics, Tome 31 (2024) no. 2. doi: 10.37236/11860

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