Problems in Extremal and Probabilistic Combinatorics

Extremal and probabilistic combinatorics is a central and currently maybe the most active and fastest growing area in discrete mathematics. The field can be traced back to the work of Turán and it was established by Erdős through his fundamental contributions and his uncounted guiding questions. Since then it has grown into an important discipline with strong ties to other mathematical areas such as theoretical computer science, number theory, and ergodic theory. The PI proposes a variety of extremal problems for hypergraphs and for sparse random and pseudorandom graphs. The work for hypergraphs is motivated by Turán’s problem, maybe the most prominent open problem in the area. After solving an analogous question for graphs, Turán asked to determine the maximum cardinality of a set E of three element subsets of a given n-element set V such that for any 4 elements of V at least one triple is missing in E. This innocent looking problem seems to be out of reach by our current methods and despite a great deal of effort over the last 70 years, our knowledge is still very limited.

We suggest a variant of the problem by imposing additional restrictions on the distribution of the three-element subsets in E. These additional assumptions yield a finer control over the corresponding extremal problem. In fact, this leads to many interesting and hopefully more manageable subproblems, some of which were already considered by Erdős and Sós. We suggest a unifying framework for these problems and one of the main goals would be the development of new techniques for this type of problems. These additional assumptions on the hyperedge distribution are closely related to the theory of quasirandom discrete structures, which was pioneered by Szemerédi and became a central theme in the field. In fact, the hypergraph extension by Gowers and by Rödl et al. of the regularity lemma provide essential tools for this line of research.

The project concerns extremal problems for uniformly dense hypergraphs, thresholds in random discrete structures, and extremal problems for sparse pseuorandom graphs. The work of the PI and his team led to 26 research articles that were submitted for publication (22 appeared, 1 accepted, 3 submitted) and we highlight one result from each of the three areas. In addition, 4 more manuscripts are close to be finished and will be submitted in 2023.

Extremal problems for uniformly dense hypergraphs
In joint work with Araújo and Piga we obtained optimal conditions for the existence of tight Hamilton cycles in 3-uniform hypergraphs that satisfy a localised pair degree condition. Problems of this type were first considered by Lenz, Mubayi, and Mycroft for loose Hamilton cycles and Aigner-Horev and Levy considered it for tight Hamilton cycles for a fairly strong notion of uniformly dense hypergraphs. We focus on tight cycles and obtain optimal results for a weaker notion of uniformly dense hypergraphs. We showed that if an n-vertex 3-uniform hypergraph H=(V,E) has the property that for any set of vertices X and for any collection P of pairs of vertices, the number of hyperedges composed by a pair belonging to P and one vertex from X is at least (1/4+c)|X||P| and H has a minimum vertex degree at least cn^2 for any fixed c>0 and sufficiently large n, then H contains a tight Hamilton cycle. A probabilistic construction shows that the constant 1/4 is optimal in this context.

Thresholds in random discrete structure
Together with Dudek, Reiher (postdoc in the project) and Ruciński we studied the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. It follows from the theorems of Dirac and of Komlós, Sarközy, and Szemerédi that for every k>0 and sufficiently large n already the minimum degree at least kn/(k+1) for an n-vertex graph G alone suffices to ensure the existence of a k-th power of a Hamiltonian cycle. We showed that under essentially the same degree assumption the addition of just Cn random edges for sufficiently large C>1 ensures the presence of the (k+1)-st power of a Hamiltonian cycle with probability close to one.

Extremal problems for sparse pseuorandom graphs
In collaboration with Berger (Ph.D. student in the project) and J. Lee (postdoc in the project) we obtained an optimal transference theorem of the Erdős-Stone theorem for odd cycles in the context of (n,d,λ)-graphs. Such a result was conjecture by Krivelevich, C. Lee, and Suakov. In addition, it was shown that edge partitions of such pseudorandom graphs into two classes leads to the expected number of odd cycles that are completely contained in one of the classes. In other words we transferred the commonness property of odd cycles in Ramsey theory into the setting of sparse pseudorandom graphs.

The project leads to a better understanding of the interplay of the different notions of quasirandomness and uniform density for hyppergraphs. This understanding seems to be crucial for extremal and Ramsey-type problems for hypergraphs. The chosen problems are central in the field and working and progressing on these problems will give the junior academic staff involved in the project a good introduction to the field and will provide a good start for their respective careers. The PI was invited to deliver a talk at the International Congress of Mathematicians in 2022, where he presented the work on extremal problems for uniformly dense hypergraphs, which is the main focus in this project.

Periodic Reporting for period 4 - PEPCo (Problems in Extremal and Probabilistic Combinatorics)

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