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On the Bollob\'as inequality
Abstract
In this thesis we are concerned with the poset ${\cal P}(n)={\cal P}(\{1,2,\...,n\} ),$ the power set of $\lbrack n\rbrack=(\{1,2,\..., n\},$ ordered by inclusion. Consider a collection of pairs of sets, $\{(A\sb{i},B\sb{i})\}\sbsp{i=1}{m}$ in ${\cal P}(n)$ with the property that $\Vert A\sb{i}\Vert=a$ and $\Vert B\sb{i}\Vert=b,$ for each $i=1,\..., m.$ The Bollobas inequality gives a bound on m in the case when $A\sb{i}\cap B\sb{j}$ is empty if and only if i = j. We present the history of the problem and its generalizations, and we make new contributions in two different directions. First we extend a result of Furedi and answer a question of Babai by giving an asymptotically sharp bound on m in the case when $\vert A\sb{i}\cap B\sb{i}\vert\le p$ for all i, and $\vert A\sb{i}\cap B\sb{j}\vert\ge q$ for all $i\ne j.$ Then we consider two set-systems ${\cal A}$ and ${\cal B}$ in the powerset ${\cal P}(n)$ with the property that for each $A\in{\cal A}$ there exists a unique $B\in{\cal B}$ such that $A\subset B.$ Ahlswede and Cai proved an inequality about such systems which is a generalization of the Bollobas inequality. We characterize the structure of the extremal cases, and exhibit a one-to-one correspondence between the extremal cases and certain matroids. ^
Subject Area
Mathematics
Recommended Citation
Szaniszlo, Zsuzsanna, "On the Bollob\'as inequality" (1996). ETD collection for University of Nebraska-Lincoln. AAI9708074.
https://digitalcommons.unl.edu/dissertations/AAI9708074