Every K_{r+1}-free graph partitions into at most (1/ε)^{C_r} parts of maximum degree at most ε|S_i|, for each r≥2.
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Sparse Partitions of Graphs with Bounded Clique Number
Every K_{r+1}-free graph partitions into at most (1/ε)^{C_r} parts of maximum degree at most ε|S_i|, for each r≥2.