# Hypergraph Turan problem

March 21 to March 25, 2011

at the

American Institute of Mathematics, San Jose, California

organized by

Dhruv Mubayi, Oleg Pikhurko, and Benny Sudakov

## Original Announcement

This workshop will be devoted to the study of the hypergraph Turan function ex(n,F), the maximum size of an F-free k-hypergraph on n vertices. Although this fundamental problem of extremal combinatorics was introduced by Paul Turan in 1941, it is still wide open in general. A number of powerful methods and techniques were developed or sharpened in recent years in order to attack various combinatorial problems (such as hypergraph regularity, flag algebras, or hypergraph stability). The purpose of the workshop is to focus this machinery on solving some imporant Turan-type questions for hypergraphs.

Two notable old problems that may be approachable by modern methods include the Tetrahedron Conjecture of Turan from 1941 and the (6,3)-problem of Ruzsa and Szemeredi from 1978.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Linear trees in uniform hypergraphs
by  Zoltan Furedi,  European J. Combin. 35 (2014), 264-272  MR3090501
Choosability with separation of complete multipartite graphs and hypergraphs
by  Zoltan Feredi, Alexandr Kostochka, and Mohit Kumbhat,  J. Graph Theory 76 (2014), no. 2, 129-137  MR3190269
Asymptotic improvements to the lower bound of certian bipartite Turán numbers
by  Simeon Ball and Valentina Pepe,  Combin. Probab. Comput. 21 (2012), no. 3, 323-329  MR2912785
On possible Turán densities
by  Oleg Pikhurko,  Israel J. Math. 201 (2014), no. 1, 415-454  MR3265290
A problem of Erdős on the minimum number of $k$-cliques
by  Shagnik Das, Hao Huang, Jie Ma, Humberto Naves, and Benny Sudakov ,  Duke Math. J. 162 (2013), no. 15, 2903-2927  MR3161307
Two extensions of Ramsey's theorem
by  David Conlon, Jacob Fox, and Benny Sudakov
On independent sets in hypergraphs
by  Alexander Kostochka, Dhruv Mubayi, and Jacques Versatraete,  Random Structures Algorithms 44 (2014), no. 2, 224-239  MR3158630
Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube
by  József Balogh, Ping Hu, Bernard Lidický, and Hong Liu,  European J. Combin. 35 (2014), 75-85  MR3090487