Pairwise Intersections and Forbidden Configurations

Abstract : Let $f_m(a,b,c,d)$ denote the maximum size of a family $\mathcal{F}$ of subsets of an $m$-element set for which there is no pair of subsets $A,B \in \mathcal{F}$ with $|A \cap B| \geq a$, $|\bar{A} \cap B| \geq b$, $|A \cap \bar{B}| \geq c$, and $|\bar{A} \cap \bar{B}| \geq d$. By symmetry we can assume $a \geq d$ and $b \geq c$. We show that $f_m(a,b,c,d)$ is $\Theta (m^{a+b-1})$ if either $b > c$ or $a,b \geq 1$. We also show that $f_m(0,b,b,0)$ is $\Theta (m^b)$ and $f_m(a,0,0,d)$ is $\Theta (m^a)$. This can be viewed as a result concerning forbidden configurations and is further evidence for a conjecture of Anstee and Sali. Our key tool is a strong stability version of the Complete Intersection Theorem of Ahlswede and Khachatrian, which is of independent interest.
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Stefan Felsner. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), pp.17-20, 2005, DMTCS Proceedings
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Richard P. Anstee, Peter Keevash. Pairwise Intersections and Forbidden Configurations. Stefan Felsner. 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), 2005, Berlin, Germany. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), pp.17-20, 2005, DMTCS Proceedings. 〈hal-01184383〉

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