Before revision:  This is obtained by combining structural properties of correlation clustering on bounded arboricity graphs with the insights of Fischer and Noever (SODA '18) on randomized greedy MIS and the   algorithm of Ailon, Charikar, and Newman (STOC '05).
After revision:  This is obtained by combining structural properties of correlation clustering on bounded arboricity graphs with the insights of Fischer and Noever (SODA '18) on randomized greedy MIS and the  PIVOT algorithm of Ailon, Charikar, and Newman (STOC '05).