Concentration and threshold phenomena in random graphs and hypergraphs

Projektinformationen

RandomHypGra

ID Finanzhilfevereinbarung: 101044123

DOI

10.3030/101044123

EK-Unterschriftsdatum 7 April 2022

Startdatum 1 Oktober 2022

Enddatum 30 September 2027

Finanziert unter

• European Research Council (ERC)

Gesamtkosten

€ 1 621 875,00

EU-Beitrag

€ 1 621 875,00

Koordiniert durch

TEL AVIV UNIVERSITY
Israel

Periodic Reporting for period 1 - RandomHypGra (Concentration and threshold phenomena in random graphs and hypergraphs)

Berichtszeitraum: 2022-10-01 bis 2025-03-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

The central theme of this research project is the emergence of 'global' structure in large random objects that are defined using a family of 'local' constraints. There are numerous well-studied instances of this abstract problem in mathematics, theoretical computer science, and mathematical physics. This research project aims to study a number of interrelated open problems spanning several branches of pure mathematics (combinatorics, Ramsey theory, and probability theory) that focus on different aspects of one unifying theme. The four main avenues of research are:

(i) The study of the global structure of a random graph conditioned on the unlikely event that the number of copies of some fixed subgraph exceeds its expectation.
(ii) The search for a general-purpose method that generalises the so-called hypergraph container lemma.
(iii) The search for an extension of the cluster expansion method in statistical physics to allow for k-wise interactions (k > 2).
(iv) Threshold phenomena associated with Turán-type and Ramsey-type problems.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

We have made substantial progress towards a range of goals set out in the proposal. The following are the most notable developments/achievements:

(i) Resolution of the Kohayakawa–Kreuter conjecture from 1997, which predicts the location of the threshold for the emergence of the asymmetric Ramsey property in the binomial random graph G(n,p) for an arbitrary sequence of graphs (H_1, ... H_r).
(ii) Determination of the sharp threshold for the exact analogue of Turán's theorem in G(n,p).
(iii) Complete description of the upper-tail deviations for the number of k-term arithmetic progressions in random subsets of integers.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

The two most significant results that go beyond the state of the art are:
(i) The development of a novel method of analysing largest and nearly-largest cuts in random graphs.
(ii) Two new proofs of the hypergraph container lemma that show unexpected connections between the container method, Janson's inequality, the Kahn–Kalai conjecture, and the hard-core model.