Objective
The central theme of this proposal is the enumeration of large structures obeying a family of local constraints. There are numerous well-studied instances of this general problem in combinatorics, probability theory, and statistical physics. We propose to study a variety of interrelated open problems, spanning extremal and probabilistic combinatorics, Ramsey theory, and large deviation theory, whose unifying theme is the distribution of copies of a given graph in large random graphs.
The first part of the proposal revolves around the 'infamous' upper tail problem for subgraph counts in random graphs, which has seen spectacular progress in the last few years. We propose to advance the combinatorial approach introduced in our recent work on a special case of the problem to solve this problem completely and make progress on several related questions.
The second part of the proposal discusses the global structure of 'sparse' subsets in hypergraphs. We propose to prove a far-reaching generalisation of the hypergraph container theorem that provides a useful description of the family of all 'sparser-than-average' sets. We plan to use it to resolve the lower tail problem for subgraph counts in the uniform random graph and to prove a sparse analogue of the counting lemma for regular graphs.
The third part of the proposal concerns the enumeration of independent sets in hypergraphs with cardinalities below the threshold for appearance of global structure. In order to obtain precise asymptotics for the number of such sets, we propose to extend the cluster expansion method from statistical physics to models with non-pairwise interactions.
The fourth part of this proposal deals with thresholds for fundamental extremal and Ramsey properties of random graphs and hypergraphs. We propose to resolve several outstanding open problems in this area that have resisted the recent dramatic advances in the field.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
- natural sciences mathematics pure mathematics discrete mathematics graph theory
- natural sciences mathematics pure mathematics discrete mathematics combinatorics
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Keywords
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
MAIN PROGRAMME
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Topic(s)
Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
HORIZON-ERC - HORIZON ERC Grants
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Call for proposal
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2021-COG
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Net EU financial contribution. The sum of money that the participant receives, deducted by the EU contribution to its linked third party. It considers the distribution of the EU financial contribution between direct beneficiaries of the project and other types of participants, like third-party participants.
69978 Tel Aviv
Israel
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.