Concentration and threshold phenomena in random graphs and hypergraphs

Informazioni relative al progetto

RandomHypGra

ID dell’accordo di sovvenzione: 101044123

DOI

10.3030/101044123

Data della firma CE 7 Aprile 2022

Data di avvio 1 Ottobre 2022

Data di completamento 30 Settembre 2027

Finanziato da

• European Research Council (ERC)

Costo totale

€ 1 621 875,00

Contributo UE

€ 1 621 875,00

Coordinato da

TEL AVIV UNIVERSITY
Israel

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

Periodo di rendicontazione: 2022-10-01 al 2025-03-31

Sintesi del contesto e degli obiettivi generali del progetto

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.

Lavoro eseguito dall’inizio del progetto fino alla fine del periodo coperto dalla relazione e principali risultati finora ottenuti

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.

Progressi oltre lo stato dell’arte e potenziale impatto previsto (incluso l’impatto socioeconomico e le implicazioni sociali più ampie del progetto fino ad ora)

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.