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Computational sciences Conferences KIAS Workshop on Combinatorics
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On Combinatorial problems of Erd?s
SPEAKER  |  Younjin Kim
INSTITUTE  |  KAIST
DATE  |  June 1(Sat), 2013
TIME  |  11:00
PLACE  |  Conference room 1503, KIAS
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ABSTRACT  |  For a property ッ and a family of sets F, let f(F,ッ) be the size of the largest subfamily of F having property ッ. For a positive integer m, let f(m,ッ) be the minimum of f(F,ッ) over all families of size m. In 1972, Erd?s and Shelah also considered ッ to be the property that no four distinct sets satisfy F1○F2=F3 and F1○F2=F4. Such families are called B2-free. Erd?s and Shelah gave an example showing f(m,B2-free)?(3/2)m2/3 and they also conjectured f(m,B2-free)>c2m2/3. We verify a conjecture of Erd?s and Shelah from 1972. In1964, Erd?s, Hajnal, and Moon introduced the following problem: get the minimum size of a graph G such that G does not contain F as a subgraph but the addition of any new edge creates at least one copy of F in G. This minimum is called the saturation number of F. We obtain the saturation number of Ck, where Ck is a cycle with length k.
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