Periodic Reporting for period 1 - RandomHypGra (Concentration and threshold phenomena in random graphs and hypergraphs)
Période du rapport: 2022-10-01 au 2025-03-31
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.
(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.
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.
(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.
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.
(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.