This project started in Fall 2018 and had to be suspended in Spring 2019 for medical reasons. At that time most of the research was still in a preliminary stage. The performed research led to 5 manuscripts (of which four appeared and one is still a preprint). All manuscripts are available on the arXiv and we briefly describe the content below:
1)Together with Glebov and Kráľ, we have done a major revision of our results on possible structure of finitely forcible graph limits, i.e. large graphs asymptotically determined by a finite number of subgraph densities. Our main result on the topic was published in 2019
Compactness and finite forcibility of graphons, Journal of the European Mathematical Society 21 (2019), 3199-3223.
2) In a joint work with Blumenthal, Lidický, Pikhurko, Pehova and Pfender, we have combined tools from the regularity method with flag algebraic techniques and solved a problem of E. Győri concerned with decomposing large graphs into cliques of small size. This work was published in 2021
Sharp bounds for decomposing graphs into edges and tringles, Combinatorics, Probability and Computing 30 (2021), 271-287.
3) In a joint work with Blumenthal, Lidický, Martin, Norin and Pfender, we coupled pseudorandom techniques with structural graph theory constructions in order to disprove a conjecture of Harris on Hall-ratio of a graph. The Hall ratio of a graph G is the maximum value of v(H)/α(H) taken over all non-null subgraphs H of G. For any graph, the Hall ratio is a lower-bound on its fractional chromatic number. We present various constructions of graphs whose fractional chromatic number grows much faster than their Hall ratio, which refutes a conjecture of Harris. This work appeared in 2022
Counterexamples to a conjecture of Harris on Hall ratio, SIAM Journal on Discrete Mathematics 36 (2022), 1678-1686.
4) Jointly with Grzesik, Lee and Lidický, we studied subgraph-density inequalities and applied them to questions concerned with so-called commonability of graphs, which appeared in 2022
On tripartite common graphs, Combinatorics, Probability and Computing 31 (2022), 907-923.
5) Together with Goaoc, Hubard, de Joannis de Verclos and Sereni, we studied quasirandom properties of point-set configurations in the plane, and their possible analytic representations analogous to limit notions in graph theory. This work was submitted in 2018
Limits of Order-types available at
https://arxiv.org/abs/1811.02236(odnośnik otworzy się w nowym oknie).
