Modern perspectives in classic extremal combinatorics

• Applicant: Professor Shagnik Das, Ph.D.
• Subject Area: Mathematics
• Term: from 2018 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 415310276

• Final Report Year: 2022

Final Report Abstract

The objective of this project was to conduct theoretical mathematical research in the field of extremal combinatorics, and to solve open problems in active areas of recent research. While the original plan had to be somewhat curtailed due to the shortened funding period and the COVID epidemic, we were able to adjust accordingly, and successfully carry out our research program primarily through local collaborations. A total of nine research articles were produced as a result of the work supported by this research grant, covering topics such as Ramsey Theory in sparse probabilistic settings, structural graph theory, combinatorial designs and combinatorial geometry. The results obtained address several open problems that have been considered by researchers in the field in recent years, and give rise to further open problems that will be the focus of future work.

Publications

• Enumerating extensions of mutually orthogonal Latin squares. Designs, Codes and Cryptography, 88(10), 2187-2206.
  Boyadzhiyska, Simona; Das, Shagnik & Szabó, Tibor
  
• Ramsey games near the critical threshold. Random Structures & Algorithms, 57(4), 940-957.
  Conlon, David; Das, Shagnik; Lee, Joonkyung & Mészáros, Tamás
  
• Vertex Ramsey properties of randomly perturbed graphs. Random Structures & Algorithms, 57(4), 983-1006.
  Das, Shagnik; Morris, Patrick & Treglown, Andrew
  
• Isomorphic bisections of cubic graphs. Journal of Combinatorial Theory, Series B, 151, 465-481.
  Das, S.; Pokrovskiy, A. & Sudakov, B.
  
• Ryser's Conjecture for t-intersecting hypergraphs. Journal of Combinatorial Theory, Series A, 179, 105366.
  Bishnoi, Anurag; Das, Shagnik; Morris, Patrick & Szabó, Tibor
  
• Subspace coverings with multiplicities. Combinatorics, Probability and Computing, 32(5), 782-795.
  Bishnoi, Anurag; Boyadzhiyska, Simona; Das, Shagnik & Mészáros, Tamás
  
• Ramsey simplicity of random graphs. Combinatorics, Probability and Computing, 34(2), 298-320.
  Boyadzhiyska, Simona; Clemens, Dennis; Das, Shagnik & Gupta, Pranshu
  
• Schur properties of randomly perturbed sets. European Journal of Combinatorics, 121, 103843.
  Das, Shagnik; Knierim, Charlotte & Morris, Patrick
  
• Tight bounds for divisible subdivisions. Journal of Combinatorial Theory, Series B, 165, 1-19.
  Das, Shagnik; Draganić, Nemanja & Steiner, Raphael