Modern perspectives in classic extremal combinatorics

• Applicant: Professor Shagnik Das, Ph.D.
• Subject Area: Mathematics
• Term: from 2018 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 415310276

Extremal combinatorics is a relatively young discipline, but one that has enjoyed tremendous growth in the past few decades, fuelled by connections to analysis, algebra, geometry, probability and theoretical computer science. The fundamental extremal problem asks how large a system can be if it does not contain any forbidden configurations. Within the subfield of extremal set theory, the classic results are the Erdős-Ko-Rado theorem and Sperner’s theorem, which determine the largest uniform intersecting families and antichains respectively.Since their discovery, these two theorems have inspired a great deal of further research. We shall study three extensions that have seen much progress in recent years. The first, Erdős-Rothschild theory, is combinatorial in nature, strengthening classic results by extending them to a multicoloured setting. The second is more probabilistic, where one seeks to determine to what extent extremal results hold in the sparse random setting. Finally, a central ingredient to many such results is the supersaturation phenomenon, which shows that many forbidden configurations must appear beyond the extremal threshold.In building upon recent work, we hope to develop general results and methods that will prove valuable to fellow researchers in the field.

• DFG Programme: Research Grants