Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, June 14, 2018, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Rajko Nenadov

On a Ramsey-Turán variant of the Hajnal-Szemerédi theorem

The Hajnal-Szemerédi theorem states that every graph G with n vertices and minimum degree at least (r-1)n/r contains a K_r-factor, that is a collection of disjoint copies of the complete graph on r vertices which together cover all the vertices of G, assuming r divides n. The bound on the minimum degree is easily seen to be optimal: take a complete r-partite graph in which one vertex class is of size n/r -1, one n/r + 1 and all other of size n/r. Such a graph has minimum degree (r-1)n/r - 1 and clearly does not contain K_r-factor. One particular feature of this example is that it contains a very large independent set. What happens if we exclude graphs with this property? Does a weaker bound on the minimum degree imply a K_r-factor if we additionally assume that G has sublinear independence number? In this talk I will give an overview of known results, present some new ones and discuss further research directions. Based on a joint work with Yanitsa Pehova (University of Warwick).

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