Concentration and threshold phenomena in random graphs and hypergraphs

Project Information

RandomHypGra

Grant agreement ID: 101044123

DOI

10.3030/101044123

EC signature date 7 April 2022

Start date 1 October 2022

End date 30 September 2027

Funded under

• European Research Council (ERC)

Total cost

€ 1 621 875,00

EU contribution

€ 1 621 875,00

Coordinated by

TEL AVIV UNIVERSITY
Israel

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

Reporting period: 2022-10-01 to 2025-03-31

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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.