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 \(r \geqslant 3\) and \(\varepsilon \in [0, 1]\) , we show that an n-vertex \((\Delta ^{\varepsilon r}, r)\) -locally sparse graph G of maximum degree \(\Delta \) satisfies \(\alpha (G) \geqslant (1-o(1))\dfrac{n}{\eta \Delta }\) and \(\chi (G) \leqslant \Theta \left( \eta \Delta \right) \) , where \(\eta :==\varepsilon + \dfrac{r\log \log \Delta }{\log \Delta }\) . For \(\varepsilon \) not too large, the hidden constant in the \(\Theta (\cdot )\) can be taken to be \(1+o(1)\) . Setting \(\varepsilon = 0\) , 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.
