Random Combinatorial Designs

Project Information

RAND-COMB-DESIGN

Grant agreement ID: 101106032

DOI

10.3030/101106032

Project closed

EC signature date 17 April 2023

Start date 1 April 2024

End date 31 March 2026

Funded under

• Marie Skłodowska-Curie Actions (MSCA)

Total cost

No data

EU contribution

€ 181 152,96

Coordinated by

UNIVERSITAT POLITECNICA DE CATALUNYA
Spain

Periodic Reporting for period 1 - RAND-COMB-DESIGN (Random Combinatorial Designs)

Reporting period: 2024-04-01 to 2026-03-31

Summary of the context and overall objectives of the project

Experiments are used across many scientific fields to test various factors, and mathematics provides the tools to ensure these tests are both fair and efficient. Randomness is employed to make experiments fair by avoiding unintended biases, while combinatorial designs (highly regular patterns like those in a completed Sudoku square) make experiments efficient by testing every variable in the most resource-effective way. Purely random layouts can be wasteful, while ordered designs may have symmetries that reintroduce bias. This project investigated random designs to capture the benefits of both approaches. The main objectives were to advance the understanding of the properties of large designs from a probabilistic perspective and to develop new mathematical pathways toward efficient ways to sample these designs. The project also explored other high-level directions in discrete mathematics, including graph bootstrap percolation, which studies how information spreads through a network, and canonical Ramsey theory in random environments, which identifies inescapable regular patterns. A secondary goal was the professional development of the researcher to ensure a successful transition to an independent research career in Europe.

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

The project was carried out through a combination of technical research, international collaboration, and academic leadership. The researcher conducted technical investigations that resulted in nine peer-reviewed publications and preprints in leading mathematics journals. These results were disseminated to the global scientific community through talks at major international congresses, such as the European Congress of Mathematics and the Random Structures and Algorithms conference, as well as through seminar visits to institutions in Brazil, Taiwan, and the United States. The researcher played a key role in animating the international research community by initiating the Barcelona Discrete Mathematics Meeting and co-organising a workshop at the Banff International Research Station. Academic mentorship was a significant part of the project, including the delivery of a specialised graduate course for the Barcelona Graduate School of Mathematics and the successful supervision of a Master’s thesis that resulted in a new research article. Public outreach was performed through popularisation articles written for general audiences to explain the impact of randomness and design theory on scientific experimentation. A major outcome of the project was the researcher successfully securing a five-year tenure-track Ramón y Cajal position at the Universitat Politècnica de Catalunya.

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 project has achieved significant progress across a diverse range of problems in discrete mathematics, delivering results that move beyond the current state of the art in several fields. In a result published in the journal Combinatorics, Probability and Computing, the researcher along with Amarja Kathapurkar and Guillem Perarnau solved a prominent case of a meta-conjecture regarding spanning structures in hypergraphs, establishing conditions under which rainbow loose Hamilton cycles must emerge in bounded colourings of dense hypergraphs. Another advance, published in the Journal of Combinatorial Designs, established a deterministic polynomial-time algorithm for finding large transversals in mathematical arrays, providing a constructive method for identifying balanced configurations. The project also established a new theoretical framework for analysing the time complexity of iterative processes on networks. This work with David Fabian and Tibor Szabó, summarised in the survey article 'Graph bootstrap percolation — a discovery of slowness' for the British Combinatorial Conference 2026, provides the mathematical community with rigorous benchmarks to calculate how long certain mathematical processes take to stabilise on large-scale networks. Furthermore, the researcher established new thresholds for regularity in random environments through canonical Ramsey theorems for even cycles and arithmetic progressions, appearing in journals such as the Proceedings of the American Mathematical Society. Finally, collaborative work with Jack Allsop on random Latin squares has generated new probability bounds that address methodological barriers previously preventing researchers from proving the efficiency of sampling methods used in bias-free experimental designs.