  We study Extremal Combinatorics problems where local properties are used to
derive global properties. That is, we consider a given configuration where
every small piece of the configuration satisfies some restriction, and use this
local property to derive global properties of the entire configuration. We
study one such Ramsey problem of Erd\H{o}s and Shelah, where the configurations
are complete graphs with colored edges and every small induced subgraph
contains many distinct colors. Our bounds for this Ramsey problem show that the
known probabilistic construction is tight in various cases. We study one
Discrete Geometry variant, also by Erd\H{o}s, where we have a set of points in
the plane such that every small subset spans many distinct distances. Finally,
we consider an Additive Combinatorics problem, where we are given sets of real
numbers such that every small subset has a large difference set.
  We derive new bounds for all of the above problems. Our proof technique is
based on introducing an non-algebraic variant of additive energies. This
abstract energy variant is based on edge colors in graphs.
