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.
