Extremal results for random discrete structures

TL;DR

This paper investigates the thresholds at which random discrete structures exhibit extremal properties, including arithmetic progressions and Turán-type problems, confirming several conjectures and extending previous results.

Contribution

It determines precise thresholds for extremal properties in random structures, including confirming a conjecture for Turán problems in random graphs and extending results to multidimensional cases.

Findings

Threshold for Szemerédi's theorem in random subsets identified

Threshold for Turán-type problems in random graphs established

Confirmed conjecture of Kohayakawa, Łuczak, and R"odl

Abstract

We study thresholds for extremal properties of random discrete structures. We determine the threshold for Szemer\'edi's theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions and we determine the threshold for Tur\'an-type problems for random graphs and hypergraphs. In particular, we verify a conjecture of Kohayakawa, \L uczak, and R\"odl for Tur\'an-type problems in random graphs. Similar results were obtained by Conlon and Gowers.