Periodic Reporting for period 1 - QuasiHyp (Quasi-random hypergraphs) Reporting period: 2018-10-01 to 2020-09-30 Summary of the context and overall objectives of the project This project aims to create novel mathematical methods that facilitate a better understanding of the properties of large networks. Large networks appear in many real-life situations such as the network of routers in the Internet, social networks like Facebook, Twitter and Linkedin, or protein-to-protein interactions inside a molecule. They can also be applied to many purely mathematical questions that, at first glance, seem completely unrelated.One of the important elements of this project will focus on providing novel ways that allow usage of computer in order to find formally correct justifications (mathematical proofs). Another important aspect of the project will revolve around randomness, which turns out to be extremely useful in various situations. In fact, for many mathematical problems involving decisions, we currently do not know anything that would perform even nearly as good as random choices do. This project aims to study ways that should improve such a situation and catch up with the randomness by being smart. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far 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 2021Sharp 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 2022Counterexamples 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 2022On 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 2018Limits of Order-types available at https://arxiv.org/abs/1811.02236. 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) As mentioned above the contributions resolve (2 and 4) or disprove (1 and 3) open problems in the field, they advance the field. Construction of a forcible graphon related to publication 1.