演講訊息 TALKS

Abstract: Extremal set theory is a subject of studying the extremal sizes or the structure of the families of subsets satisfying the given properties. We are interested in how large a family of subsets of the set [n] = {1,2,…, n} that avoids a given partially ordered set (poset) P. The foundational result of this sort, Sperner's Theorem (1928), solves this problem for families that contain no two-element chain: A B (that is, for antichains), determining that the maximum size is the binomial coefficient . Let La (n,P) be the largest size of families of subsets of [n] that do not contain the poset P. A conjecture posed by Griggs and Lu (2009), independently by Bukh (2009), is that La (n,P) is asymptotically equal to e , where e is an integer determined by P when n is large enough. Although it is generally believed to be true, the conjecture is still open even for the poset on four elements. We will see how to use a combinatorial counting method to determine the values of La (n,P) for some types of posets P, which consequently proves that the conjecture holds for those posets.