  In a fractional coloring, vertices of a graph are assigned measurable subsets
of the real line and adjacent vertices receive disjoint subsets; the fractional
chromatic number of a graph is at most $k$ if it has a fractional coloring in
which each vertex receives a subset of $[0, 1]$ of measure at least $1/k$. We
introduce and develop the theory of "fractional colorings with local demands"
wherein each vertex "demands" a certain amount of color that is determined by
local parameters such as its degree or the clique number of its neighborhood.
This framework provides the natural setting in which to generalize
degree-sequence type bounds on the independence number. Indeed, by Linear
Programming Duality, all of the problems we study have an equivalent
formulation as a problem concerning weighted independence numbers, and they
often imply new bounds on the independence number.
  Our results and conjectures are inspired by many of the most classical
results and important open problems concerning the independence number and the
chromatic number, often simultaneously. We conjecture a local strengthening of
both Shearer's bound on the independence number of triangle-free graphs and the
fractional relaxation of Molloy's recent bound on their chromatic number, as
well as a longstanding problem of Ajtai et al.\ on the independence number of
$K_r$-free graphs and the fractional relaxations of Reed's $\omega, \Delta,
\chi$ Conjecture and the Total Coloring Conjecture. We prove an approximate
version of the first two, and we prove "local demands" versions of Vizing's
Theorem and of some $\chi$-boundedness results.
