Regularity inheritance in pseudorandom graphs
  Advancing the sparse regularity method, we prove one-sided and two-sided regularity inheritance lemmas for subgraphs of bijumbled graphs, improving on results of Conlon, Fox and Zhao [Adv. Math. 256 (2014), 206--290]. These inheritance lemmas also imply improved $H$-counting lemmas for subgraphs of bijumbled graphs, for some $H$. 
Peter Allen, Julia B\"ottcher, Jozef Skokan and Maya Stein
math.CO