Configurations in sets big and small
Malabika Pramanik · Open Scholarship Institutional Repository (Washington University in St. Louis) · 2016
When does a given set contain a copy of your favorite pattern (for example, specially arranged points on a line or spiral, or the vertices of a polyhedron)? Does the answer depend on how thin the set is in some quantifiable sense? Problems involving identification of prescribed configurations under varying interpretations of size have been vigorously pursued both in the discrete and continuous setting, often with spectacular results that run contrary to intuition. Yet many deceptively simple questions remain open. I will survey the literature in this area, emphasizing some of the landmark results that focus on different aspects of the problem.