A family of subsets of {1,2,…,n} is called maximal k-wise intersecting if every collection of at most k members from has a common element, and moreover, no set can be added to while preserving this property. In 1974, Erdős and Kleitman asked for the smallest possible size of a …
A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected 2-complex every link graph of which is 3-connected admits an essentially unique locally flat embedding into the 3-sphere, if it admits one at all. This can be thought of …
The poset Ramsey number is the smallest integer such that any blue-red coloring of the elements of the Boolean lattice has a blue induced copy of or a red induced copy of . Axenovich and Walzer showed that . Recently, Lu and Thompson improved the upper bound to . In …
What is the largest number where every graph with average degree at least contains a subdivision of ? Mader asked this question in 1967 and was proved by Bollobás and Thomason and independently by Komlós and Szemerédi. This is best possible by considering a disjoint union of . However, this …
