Abstract
For a set G and a family of sets \({\mathcal{F}}\) let \({\mathcal{D}_{\mathcal{F}}(G)=\{F\in \mathcal{F}:F\cap G=\emptyset\}}\) and \({\mathcal{S}_{\mathcal{F}}(G)=\{F\in\mathcal{F}:F\subseteq G\,{\rm or} \,G \subseteq F\}.}\) We say that a family is l-almost intersecting, (≤ l)-almost intersecting, l-almost Sperner, (≤ l)-almost Sperner if \({|\mathcal{D}_{\mathcal{F}}(F)|=l, |\mathcal{D}_{\mathcal{F}}(F)|\le l, |\mathcal{S}_{\mathcal{F}}(F)|=l, |\mathcal{S}_{\mathcal{F}}(F)| \le l}\) (respectively) for all \({F \in \mathcal{F}.}\) We consider the problem of finding the largest possible family for each of the above properties. We also address the analogous generalization of cross-intersecting and cross-Sperner families.
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Research of Dániel Gerbner, Nathan Lemons, Cory Palmer was supported by Hungarian National Scientific Fund, grant number: OTKA NK-78439.
Research of Balázs Patkós was supported by Hungarian National Scientific Fund, grant numbers: OTKA K-69062 and PD-83586 and the János Bolyai Reasearch Scholarship of the Hungarian Academy of Sciences.
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Gerbner, D., Lemons, N., Palmer, C. et al. Almost Cross-Intersecting and Almost Cross-Sperner Pairs of Families of Sets. Graphs and Combinatorics 29, 489–498 (2013). https://doi.org/10.1007/s00373-012-1138-2
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DOI: https://doi.org/10.1007/s00373-012-1138-2