Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Occupancy fraction, fractional colouring, and triangle fraction   Given $\varepsilon>0$, there exists $f_0$ such that, if $f_0 \le f \le
\Delta^2+1$, then for any graph $G$ on $n$ vertices of maximum degree $\Delta$
in which the neighbourhood of every vertex in $G$ spans at most $\Delta^2/f$
edges, (i) an independent set of $G$ drawn uniformly at random has at least
$(1/2-\varepsilon)(n/\Delta)\log f$ vertices in expectation, and (ii) the
fractional chromatic number of $G$ is at most $(2+\varepsilon)\Delta/\log f$.
These bounds cannot in general be improved by more than a factor $2$
asymptotically. One may view these as stronger versions of results of Ajtai,
Koml\'os and Szemer\'edi (1981) and Shearer (1983). The proofs use a tight
analysis of the hard-core model.

23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
3
The categories of this paper are: physics
16
The categories of this paper are: stat
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
11
The categories of this paper are: hep-lat