A Ramsey–Turán theory for tilings in graphs

For a (Formula presented.) -vertex graph (Formula presented.) and an (Formula presented.) -vertex graph (Formula presented.), an (Formula presented.) -tiling in (Formula presented.) is a collection of vertex-disjoint copies of (Formula presented.) in (Formula presented.). For (Formula presented.), the (Formula presented.) -independence number of (Formula presented.), denoted (Formula presented.), is the largest size of a (Formula presented.) -free set of vertices in (Formula presented.). In this article, we discuss Ramsey–Turán-type theorems for tilings where one is interested in minimum degree and independence number conditions (and the interaction between the two) that guarantee the existence of optimal (Formula presented.) -tilings. Our results unify and generalise previous results of Balogh–Molla–Sharifzadeh [Random Struct. Algoritm. 49 (2016), no. 4, 669–693], Nenadov–Pehova [SIAM J. Discret. Math. 34 (2020), no. 2, 1001–1010] and Balogh–McDowell–Molla–Mycroft [Comb. Probab. Comput. 27 (2018), no. 4, 449–474] on the subject.