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Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569--584] proved that any set of simple closed curves on the torus, where any two are non-homotopic and intersect at most $k$ times, has a maximum size of $k+O(\sqrt{k}\log k)$. We prove the maximum size of such a set is at most $k+O(1)$ and determine the exact maximum size for all sufficiently large $k$. In particular, we show that the maximum does not exceed $k+4$ when $k$ is large., Comment: 22 pages

A classical Tur\'an problem asks for the maximum possible number of edges in a graph of a given order that does not contain a particular graph $H$ as a subgraph. It is well-known that the chromatic number of $H$ is the graph parameter which describes the asymptotic behavior of this maximum. Here, we consider an analogous problem for oriented graphs, where compressibility plays the role of the chromatic number. Since any oriented graph having a directed cycle is not contained in any transitive tournament, it makes sense to consider only acyclic oriented graphs as forbidden subgraphs. We provide basic properties of the compressibility, show that the compressibility of acyclic oriented graphs with out-degree at most 2 is polynomial with respect to the maximum length of a directed path, and that the same holds for a larger out-degree bound if the Erd\H{o}s-Hajnal conjecture is true. Additionally, generalizing previous results on powers of paths and arbitrary orientations of cycles, we determine the compressibility of acyclic oriented graphs with a restricted structure., Comment: 14 pages

An oriented graph $H$ is quasirandom-forcing if the limit (homomorphism) density of $H$ in a sequence of tournaments is $2^{-\|H\|}$ if and only if the sequence is quasirandom. We study generalizations of the following result: the cyclic orientation of a cycle of length $\ell$ is quasirandom-forcing if and only if $\ell\equiv 2$ mod $4$. We show that no orientation of an odd cycle is quasirandom-forcing. In the case of even cycles, we find sufficient conditions on an orientation to be quasirandom-forcing, which we complement by identifying necessary conditions. Using our general results and spectral techniques used to obtain them, we classify which orientations of cycles of length up to $10$ are quasirandom-forcing.

We consider the problem of determining the inducibility (maximum possible asymptotic density of induced copies) of oriented graphs on four vertices. We provide exact values for more than half of the graphs, and very close lower and upper bounds for all the remaining ones. It occurs that, for some graphs, the structure of extremal constructions maximizing density of its induced copies is very sophisticated and complex., Comment: Added appendices on usage of updated codes and verification of the certificates

Alon and F\"{u}redi (1993) showed that the number of hyperplanes required to cover $\{0,1\}^n\setminus \{0\}$ without covering $0$ is $n$. We initiate the study of such exact hyperplane covers of the hypercube for other subsets of the hypercube. In particular, we provide exact solutions for covering $\{0,1\}^n$ while missing up to four points and give asymptotic bounds in the general case. Several interesting questions are left open., Comment: Small fixes and updated code

We prove that for each odd integer $k \geq 7$, every graph on $n$ vertices without odd cycles of length less than $k$ contains at most $(n/k)^k$ cycles of length $k$. This generalizes the previous results on the maximum number of pentagons in triangle-free graphs, conjectured by Erd\H{o}s in 1984, and asymptotically determines the generalized Tur\'an number $\mathrm{ex}(n,C_k,C_{k-2})$ for odd $k$. In contrary to the previous results on the pentagon case, our proof is not computer-assisted.