Christian Reiher

The clique density theorem

Tur an’s theorem is a cornerstone of extremal graph theory. It asserts that for any integer r > 2, every graph on n vertices with more than r 2 2(r 1) n 2 edges contains a clique of sizer, i.e.,r mutually adjacent vertices. The corresponding extremal graphs are balanced (r 1)-partite graphs. The question as to how many suchr-cliques appear at least in anyn-vertex graph with n 2 edges has been intensively studied in the literature. In particular, Lov asz and Simonovits conjectured in the 1970’s that asymptotically the best possible lower bound is given by the complete multipartite graph with n 2 edges in which all but one vertex class is of the same size while the remaining one may be smaller. Their conjecture was recently resolved for r = 3 by Razborov and for r = 4 by Nikiforov. In this article, we prove the conjecture for all values ofr.

Embedding tetrahedra into quasirandom hypergraphs

We investigate extremal problems for quasirandom hypergraphs. We say that a

A Ramsey class for Steiner systems

3

Hypergraphs with vanishing Turán density in uniformly dense hypergraphs

-uniform hypergraph

The chromatic number of finite type-graphs

H=(V,E)

Notes: Counting odd cycles in locally dense graphs

is

Maximum star densities

(d,\eta)

A tale of stars and cliques

-quasirandom if for any subset

NASH-WILLIAMS’ THEOREM ON DECOMPOSING GRAPHS INTO FORESTS

X\subseteq V

On a Turán problem in weakly quasirandom 3-uniform hypergraphs

and every set of pairs