Bounds for the independence and chromatic numbers of locally sparse graphs

In this note we consider a more general version of local sparsity introduced recently by Anderson, Kuchukova, and the author. In particular, we say a graph $G = (V, E)$ is $(k, r)$-locally-sparse if for each vertex $v \in V(G)$, the subgraph induced by its neighborhood contains at most $k$ cliques of size $r$. For $r \geq 3$ and $\epsilon \in [0, 1]$, we show that an $n$-vertex $(\Delta^{\epsilon r}, r)$-locally-sparse graph $G$ of maximum degree $\Delta$ satisfies $\alpha(G) \geq (1-o(1))\dfrac{n}{\eta\Delta}$ and $\chi(G) \leq \Theta\left(\eta\Delta\right)$, where $\eta := \epsilon + \dfrac{r\log\log \Delta}{\log \Delta}$. For $\epsilon$ not too large, the hidden constant in the $\Theta(\cdot)$ can be taken to be $1+o(1)$. Setting $\epsilon = 0$, we recover classical results on $K_{r+1}$-free graphs due to Shearer and Johansson, which were more recently improved by Davies, Kang, Pirot, and Sereni. We prove a stronger result on the independence number in terms of the occupancy fraction in the hard-core model, and establish a local version of the coloring result in the more general setting of correspondence coloring.