On the size of the largest P -free families

We will study the problem of determining the maximum size of a family of subsets of [n] = {1, 2, . . . , n} not containing a given poset P as a (weak) subposet, denoted La(n, P ). This problem is a generalization of the well-known Sperner’s theorem. In 1945, Erdős obtained the exact value of La(n, P ) when P is a path poset, generalizing Sperner’s theorem. A more formal study of this problem was initiated by Katona and Tarján in 1983. Since then there have been numerous papers in this area and many open questions. One of the open questions was to obtain a good general bound on La(n, P ) for an arbitrary poset P . Open questions concerning the exact (or at least asymptotic) value of La(n, P ) for some specific posets P are also of great interest. The most famous poset of which, is the Diamond. In this thesis, we answer some of these open questions. We obtain a general bound on La(n, P ) which is best possible upto a constant factor, improving the previous bounds due to Burcsi and Nagy and later Chen and Li. We also obtain the exact value of La(n, P ) for an infinite class of posets and introduce a new method for doing so. The thesis consists of 3 chapters: In the first chapter we survey results about forbidden subposets and prove some well-known theorems. In the second chapter we show La(n, P ) ≤ 1 2k−1 ( |P |+ (3k − 5)2k−2(h(P )− 1)− 1 ) ( n bn/2c ) for any fixed integer k ≥ 2, improving the best known upper bound. By choosing k appropriately, we obtain that La(n, P ) = O ( h(P ) log2 ( |P | h(P ) + 2 )) ( n bn/2c ) as a corollary, which we show is best possible for general P . We also give a different proof of this corollary by using bounds for generalized diamonds. We also show that the Lubell function of a family of subsets of [n] not containing P as an induced subposet is O(n) for every c > 1 2 . This is joint work with Dániel Grósz and Casey Tompkins. In the third chapter, we introduce a method of decomposing the family of intervals along a cyclic permutation into chains to determine the exact size of the largest family of subsets of [n] not containing one or more given posets as a subposet. De Bonis, Katona and Swanepoel determined the size of the largest butterfly-free family. We strengthen this result by showing that, for certain posets containing the butterfly poset as a subposet, the same bound holds. We also obtain the corresponding LYM-type inequalities. This is joint work with Casey Tompkins.