Eoin Long

Long paths and cycles in subgraphs of the cube

Let Qn denote the graph of the n-dimensional cube with vertex set {0, 1}n in which two vertices are adjacent if they differ in exactly one coordinate. Suppose G is a subgraph of Qn with average degree at least d. How long a path can we guarantee to find in G?Our aim in this paper is to show that G must contain an exponentially long path. In fact, we show that if G has minimum degree at least d then G must contain a path of length 2d − 1. Note that this bound is tight, as shown by a d-dimensional subcube of Qn. We also obtain the slightly stronger result that G must contain a cycle of length at least 2d.

A stability result for the cube edge isoperimetric inequality

Abstract We prove the following stability version of the edge isoperimetric inequality for the cube: any subset of the cube with average boundary degree within K of the minimum possible is e-close to a union of L disjoint cubes, where L ≤ L ( K , e ) is independent of the dimension. This extends a stability result of Ellis, and can viewed as a dimension-free version of Friedguts junta theorem.

Forbidding intersection patterns between layers of the cube

A family A ? P n ] is said to be an antichain if A ? B for all distinct A , B ? A . A classic result of Sperner shows that such families satisfy | A | ? ( n ? n / 2 ? ) , which is easily seen to be best possible. One can view the antichain condition as a restriction on the intersection sizes between sets in different layers of P n ] . More generally one can ask, given a collection of intersection restrictions between the layers, how large can families respecting these restrictions be? Answering a question of Kalai 8], we show that for most collections of such restrictions, layered families are asymptotically largest. This extends results of Leader and the author from 11].

Set families with a forbidden pattern

A balanced pattern of order

Packing, Counting and Covering Hamilton cycles in random directed graphs

2d

Packing and counting arbitrary Hamilton cycles in random digraphs

is an element

Large Unavoidable Subtournaments

P \in \{+,-\}^{2d}

Large unavoidable subtournaments

, where both signs appear

Frankl-Rödl-type theorems for codes and permutations

d

Tilted Sperner families

times. Two sets