  A graph $G$ is $k$-locally sparse if for each vertex $v \in V(G)$, the
subgraph induced by its neighborhood contains at most $k$ edges. Alon,
Krivelevich, and Sudakov showed that for $f > 0$ if a graph $G$ of maximum
degree $\Delta$ is $\Delta^2/f$-locally-sparse, then $\chi(G) =
O\left(\Delta/\log f\right)$. We introduce a more general notion of local
sparsity by defining graphs $G$ to be $(k, F)$-locally-sparse for some graph
$F$ if for each vertex $v \in V(G)$ the subgraph induced by the neighborhood of
$v$ contains at most $k$ copies of $F$. Employing the R\"{o}dl nibble method,
we prove the following generalization of the above result: for every bipartite
graph $F$, if $G$ is $(k, F)$-locally-sparse, then $\chi(G) = O\left( \Delta
/\log\left(\Delta k^{-1/|V(F)|}\right)\right)$. This improves upon results of
Davies, Kang, Pirot, and Sereni who consider the case when $F$ is a path. Our
results also recover the best known bound on $\chi(G)$ when $G$ is $K_{1, t,
t}$-free for $t \geq 4$, and hold for list and correspondence coloring in the
more general so-called ''color-degree'' setting.
