Extremal Problems on Finite Sets and Posets

The overarching theme of the thesis is the investigation of extremal problems involving forbidden partially ordered sets (posets). In particular, we will be concerned with the function La(n, P ), defined to be the maximum number of sets we can take in the Boolean lattice 2[n] without introducing the relations of a poset P as containment relations among the sets. This function plays an analogous role in the setting of nonuniform hypergraphs (set systems) as the extremal function ex(n,G) in graph theory and has already been studied extensively. While this type of problem was formally introduced in 1983, there have already been more than 50 papers on the subject. The majority of the results involve bounding La(n, P ) both in a general sense, as well as for numerous specific cases of posets. The thesis is divided into 5 main chapters. The first chapter gives a summary of the history of forbidden poset problems as well as the relevant background information from extremal set theory. In the second chapter we give a significant improvement of the general bounds on La(n, P ) as a function of the height of the poset h(P ) and the size of the poset |P | due to Burcsi and Nagy and later Chen and Li. The resulting bound is in a certain sense best possible. We also give an improvement of the bound on the so-called Lubell function in the induced version of the problem, and we introduce a new chain counting technique which may have additional applications. The results in this chapter were joint work with Dániel Grósz and Abhishek Methuku. The third chapter introduces a new partitioning technique on cyclic permutations. In this chapter we find a surprising generalization of a result of De Bonis, Katona and Swanepoel on a poset known as the butterfly. Namely, we show that one can introduce a subdivision to one of the edges of the Hasse diagram of this poset and prove that nonetheless the same bound holds on its extremal number. Using the new partitioning technique, we also determine the exact bound for the La function of an infinite class of pairs of posets. The results in this chapter were joint work with Abhishek Methuku. In the fourth chapter we introduce a new variation on the extremal subposet problem. Namely, we assume that the family must also be intersecting. We give a novel generalization of the partition method of Griggs and Li and prove an exact bound for intersecting, butterfly-free families. We also give a new proof of a result of Gerbner on intersecting k-Sperner families and determine the equality cases for the first time. The results in this chapter were joint work with Dániel Gerbner and Abhishek Methuku. In the fifth chapter we prove a new version of the De Bruijn Erdős theorem for partially ordered sets, as well as another version in a graph setting. The results in this subsection were joint work with Pierre Aboulker, Guillaume Lagarde, David Malec and Abhishek Methuku.