Ramsey–Turán theory Original Research Article

Ramsey- and Turán-type problems were always strongly related to each other. Motivated by an observation of Paul Erdős, it was Turán who started the systematic investigation of the applications of extremal graph theory in geometry and analysis. This led the second author to some results and problems which, in turn, led to the birth of Ramsey–Turán-type theorems. Today this is a wide field of research with many interesting results and many unsolved problems. Below we give a short survey of the most important parts of this field: starting with a historical sketch we continue by describing the 1. Ramsey–Turán-type problems and results. 2. Related problems in Ramsey theory. 3. Some applications.