Andrásfai and Vega graphs in Ramsey–Turán theory

Abstract

Given positive integers ⩾ n s, we let n s ex( , ) denote the This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

Geometric constructions for Ramsey-Turán theory

“…Mantel's theorem and an early result of Andrásfai [3] determine ex 3 (ε) for ε ≥ 2 5 ; and the regime ε ∈ (0, 1 3 ] follows from work of Brandt [11]. Very recently, Luczak, Polcyn and Reiher determined ex 3 (ε) for ε ∈ [ 3 8 , 2 5 ] [23] and certain intervals in the remaining range ( 13 , 3 8 ) [24] where they have a conjecture. Determining ex q (n, m) for functions m(n) growing slower than linear has also attracted a lot of attention, in particular determining the phase transitions where decreasing m(n) causes a large decrease in ex q (n, m) (for a precise definition see [4]).…”