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This part is connected to a problem of Erdős and Turán from the 1930s: related to the van der Waerden theorem, they asked if the density version of that result also holds:

Is it true that an infinite sequence of integers of positive (lower) density contains arbitrarily long arithmetic progressions?

We are going to give exact bound for the size of longest arithmetic progression in sumset sums. In addition, we describe the structure of the subset sums, and give applications in number theory and probability theory. (This part is partially joint work with Van Vu.)

I will describe and illustrate a method to embed relatively sparse graphs into large graphs. This will include the case of Pósa's conjecture, El Zahar's conjecture, and tree embedding under different conditions. Among other things, we shall give several generalizations of the central Dirac Theorem, both for graphs and hypergraphs. The methods used are elementary. A large part of this work is joint with coauthors, e.g. Asif Jamsed, Imdadullah Khan, Sarmad Abbasi, and Gábor Sárközy.

Endre Szemerédi

Endre Szemerédi was born in Hungary and received his PhD from Moscow University in 1970 under the direction of I.M. Gelfand. His research interests include arithmetic combinatorics, extremal graph theory, elementary number theory and theoretical computer science. Since 1986, he has held a State of New Jersey Professorship of Computer Science at Rutgers University, and he is also a permanent research fellow at Alfréd Rényi Mathematical Institute. Professor Szemerédi has received numerous prizes and honors, including being elected to membership of the Hungarian Academy of Sciences in 1989, and to the American National Academy of Sciences in 2010. He is married and has five children.

DeLong Lecture Series