The first author together with Jenssen, Perkins and Roberts (2017) recently showed how
local properties of the hard-core model on triangle-free graphs guarantee the existence of …

The problem of finding dense induced bipartite subgraphs in H-free graphs has a long
history, and was posed 30 years ago by Erdős, Faudree, Pach and Spencer. In this paper …

We study a restricted form of list colouring, for which every pair of lists that correspond to
adjacent vertices may not share more than one colour. The optimal list size such that a …

In 1992, Erd\H {o} s and Hajnal posed the following natural problem: Does there exist, for
every $ r\in\mathbb {N} $, an integer $ F (r) $ such that every graph with chromatic number at …

The Hall ratio of a graph G is the maximum of| V (H)|/α (H) over all subgraphs H of G. It is
easy to see that the Hall ratio of a graph is a lower bound for the fractional chromatic …

A graph is $ H $-Ramsey if every two-coloring of its edges contains a monochromatic copy
of $ H $. Define the $ F $-Ramsey number of $ H $, denoted by $ r_F (H) $, to be the …

We prove the following local strengthening of Shearer's classic bound on the independence
number of triangle-free graphs: For every triangle-free graph $ G $ there exists a probability …

The Hall ratio of a graph G is the maximum value of v(H)/α(H) taken over all non-null
subgraphs H⊆G. For any graph, the Hall ratio is a lower-bound on its fractional chromatic …

A well-known conjecture by Harris states that any triangle-free $ d $-degenerate graph has
fractional chromatic number at most $ O\left (\frac {d}{\ln d}\right) $. This conjecture has …

A bull is the graph obtained by adding two pendant edges at different vertices of a triangle. A
diamond is the graph obtained from a $ K_4 $ by deleting an edge. In this paper, we study …