Delta-system method: a survey

Abstract

In 1960 Erd\H os and Rado published a paper that, in retrospect, became one of the most influential papers in extremal set theory. They proved a result of Ramsey theoretic flavour, stating that in any sufficiently large family of sets of bounded size there is a homogeneous substructure, called a $\Delta$-system (also known under the name of a sunflower). For many qualitative results in Discrete Mathematics and Theoretical Computer Science, this has become a very powerful tool to analyze complex set families. Extremal set theory flourished in the 1970's--80's, and many exciting developments happened then. One of them was the development of the $\Delta$-system method in the works of Frankl and F\"uredi. In this survey, we try to give a concise picture of this method starting from its early stages and to the modern day. We also tried to present the proofs of most of the key results. On top of this, we survey the literature on the problems that the Delta-systems was applied to.