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Abstract: We describe a simple and yet surprisingly powerful probabilistic techniquewhich shows how to find in a dense graph a large subset of vertices in whichall or almost all small subsets have many common neighbors. Recently thistechnique has had several striking applications to Extremal Graph Theory,Ramsey Theory, Additive Combinatorics, and Combinatorial Geometry. In thissurvey we discuss some of them.

Author: Jacob Fox, Benny Sudakov


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