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Thursday, October 1, 2015

**Abstract:** Cameron and Erdős proposed several of enumeration problems in additive combinatorics, for example what is the number of sum-free sets in [n], or what is the number of sets in [n] which do not contain k-term arithmetic progression. I plan to survey the recent progress on these type of questions, focusing on the applications of a recent powerful tool, the hyper graph container lemma of Balogh-Morris-Samotij, and of Saxton-Thomason. It is joint work with H. Liu, M. Sharifzadeh, and A. Treglown.