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Almost Cross-Intersecting and Almost Cross-Sperner Pairs of Families of Sets |
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| Authors: | Dániel Gerbner Nathan Lemons Cory Palmer Dömötör Pálvölgyi Balázs Patkós Vajk Szécsi |
| Institution: | 1. Hungarian Academy of Sciences, Alfréd Rényi Institute of Mathematics, P.O.B. 127, Budapest, 1364, Hungary 2. Theoretical Division, Los Alamos National Laboratory, LosAlamos, NM, 87545, USA 3. Department of Mathematics, University of Illinois at Urbbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801, USA 4. Department of Computer Science Pázmány Péter sétány1/C, E?tv?s Loránd University, Budapest, 1117, Hungary 5. Department of Mathematics and its Applications, Central European University, Nádor u. 9, Budapest, 1051, Hungary |
| 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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