Two sets are non-crossing if they are disjoint or one contains the other. The noncrossing graph NCn is the graph whose vertex set is the set of nonempty subsets of [n] = {1, . . . , n} with an edge between any two non-crossing sets. Various facts, some new and some already known, concerning the chromatic number, fractional chromatic number, independence number, clique number and clique cover nu...

For a graph G, a random n-lift of G has the vertex set V (G) n] and for each edge u; v] 2 E(G), there is a random matching between fug n] and fvg n]. We present bounds on the chromatic number and on the independence number of typical random lifts, with G xed and n ! 1. For the independence number, upper and lower bounds are obtained as solutions to certain optimization problems on the base grap...

The measurable list chromatic number of a graph G is the smallest number ξ such that if each vertex v of G is assigned a set L(v) of measure ξ in a fixed atomless measure space, then there exist sets c(v) ⊆ L(v) such that each c(v) has measure one and c(v) ∩ c(v′) = ∅ for every pair of adjacent vertices v and v′. We provide a simpler proof of a measurable generalization of Hall’s theorem due to...

We prove a tight connection between two important notions in combinatorial optimization. Let G be a graph class (i.e. a subset of all graphs) and r(G) = supG∈G χf (G) ω(G) where χf (G) and ω(G) are the fractional chromatic number and clique number of G respectively. In this note, we prove that r(G) tightly captures the integrality gap of the LP relaxation with clique constraints for the Maximum...