SYGN 2012 Abstracts - Tamas Szonyi

Stability theorems in combinatorics and finite geometry
Tamas Szonyi (Eötvös Loránd University, Hungary)

A stability theorem says that an `almost nice' structure can be obtained from a `nice structure' by a
`small' modification. Here a structure can be `nice' if it has a large automorphism group, or it is very regular in the combinatorial sense, or it is simply extremal regarding the value of a certain (numerical) parameter. Of course, the key problem is to formulate precisely what `almost nice' and `small modification' mean. The prototype of such result is the stability theorem for the Tur\'an-graphs, due to Erd\{o}s and Simonovits. In finite geometry, the results of B. Segre on embeddability of large arcs in (hyper)ovals are typical examples. The aim of the talk is to survey old and new stability results in various branches of discrete mathematics.