Progress on Poset-free Families of Subsets

Increasing attention is being paid to the study of families of subsets of an nset that contain no subposet P . Especially, we are interested in such families of maximum size given P and n. For certain P this problem is solved for general n, while for other P it is extremely challenging to find even an approximate solution for large n. It is conjectured that for any P , the maximum size is asymptotic to a constant times ( n b 2 c ) , where the constant is a certain integer depending on P . This survey has two purposes. First, we want to bring this exciting line of research to the attention of a wider audience. Second, we want to make experts aware of the broad range of recent progress in the area. 1 First Results: Families Without Chains Problems of forbidding given structures in a larger structure are very popular in extremal combinatorics. The study of forbidden subgraph problems, starting from Mantel’s result on triangle-free graphs, is nowadays a well-developed discipline. The Turán theory on hypergraphs contains beautiful theorems and challenging unsolved problems. A research area that has become fertile in recent years considers forbidding poset structures in families of subsets. Let P = (P,≤) be a finite poset. We say that another poset Q contains P as a subposet (in the weak sense), if there exists an order-preserving injection i from P to Q. We view a family F of subsets of a finite set as a poset itself, ordered according to the inclusion relation of sets. More precisely, working in the Boolean lattice Bn of all 2 subsets of [n] := {1, 2, · · · , n}, ordered by inclusion, we want to understand how large a family of subsets in Bn can be without containing a given poset P as a subposet. We let ( [n] k ) denote the collection of k-subsets of [n]. ∗Department of Mathematics, University of South Carolina, Columbia, SC, USA 29208 ([email protected]). Research supported in part by a grant from the Simons Foundation (#282896 to Jerrold Griggs) and by a long-term visiting position at the IMA, University of Minnesota. †Department of Applied Mathematics, National Chung Hsing University, Taichung, Taiwan 40227 ([email protected]). Supported by Ministry of Science and Technology (No. 103-2115-M-005 -003 -MY2)