In several classical areas of combinatorics, the underlying structure itself is equipped with some partial or total order. For example, the subsets of a set are naturally ordered by containment. A typical question in extremal set theory is the following Sperner-type problem: given a set X, what is the maximum number of subsets we can select without including two that are comparable, or without forming some other forbidden substructure in the Boolean lattice of subsets of X? The theory of partially ordered sets was developed to address such problems, and remains a vital area within the field of combinatorics.
The main goal of the thematic semester is to study and advance powerful new techniques related to ordered structures. Many recent breakthroughs in combinatorics implement such methods or can be expected
to have further highly significant extensions to an ordered setting. Examples include Erdos-Hajnal conjecture for hereditary graph classes (Scott-Seymour), New bounds on Ramsey numbers (Campos-Griffiths-Morris-Sahasrabudhe, Mattheus-Verstraete), Euclidean Ramsey theory (de Grey), Marton's conjecture, Freiman's theorem (Ruzsa, Gowers-Tao), Extremal matrix theory with forbidden 0-1 matrices (Pettie-Tardos), The finite field and 3-dimensional Kakeya conjectures (Dvir, Wang-Zahl), Orchard planting problem (Green–Tao).