MATH Seminar
| Title: On Erdos-Ko-Rado-type theorems |
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| Colloquium: N/A |
| Speaker: Peter Frankl of The Hungarian Academy of Sciences |
| Contact: Dwight Duffus, dwight@mathcs.emory.edu |
| Date: 2013-05-01 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: The lecture is going to focus on extremal set theory. The general problem is concerned with the maximum possible size of a subset of the power set of a finite set $X$ of $n$ elements subject to some conditions. The simplest result is probably the following.\\ \\ Proposition 0. If $F$ is a subset of $2^X$, such that any two sets in $F$ have non-empty intersection then $|F| \leq 2^(n-1)$.\\ \\ One way to achieve equality is by taking all subsets containing a fixed element.\\ \\ Erdös-Ko-Rado Theorem. If $F$ is a collection of $k$-element subsets of $X$ such that any two sets in $F$ have non-empty intersection and $2k < n$ , then $|F| \leq {n-1 \choose k-1}$ with equality holding only if all subsets in $F$ contain a fixed element. We are going to discuss various generalizations and extensions of this result, some of which are still unsolved. |
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