Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe extremal number of surfaces
72
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In 1973, Brown, ErdH{o}s and Sos proved that if $mathcal{H}$ is a 3-uniform hypergraph on $n$ vertices which contains no triangulation of the sphere, then $mathcal{H}$ has at most $O(n^{5/2})$ edges, and this bound is the best possible up to a constant factor. Resolving a conjecture of Linial, also reiterated by Keevash, Long, Narayanan, and Scott, we show that the same result holds for triangulations of the torus. Furthermore, we extend our result to every closed orientable surface $mathcal{S}$.
rate research
Read More
We show that there exists an absolute constant $C>0$ such that any family $mathcal{F}subset {0,1}^n$ of size at least $Cn^3$ has dual VC-dimension at least 3. Equivalently, every family of size at least $Cn^3$ contains three sets such that all eight regions of their Venn diagram are non-empty. This improves upon the $Cn^{3.75}$ bound of Gupta, Lee and Li and is sharp up to the value of the constant.
In this paper, we give bounds on the dichromatic number $vec{chi}(Sigma)$ of a surface $Sigma$, which is the maximum dichromatic number of an oriented graph embeddable on $Sigma$. We determine the asymptotic behaviour of $vec{chi}(Sigma)$ by showing that there exist constants $a_1$ and $a_2$ such that, $ a_1frac{sqrt{-c}}{log(-c)} leq vec{chi}(Sigma) leq a_2 frac{sqrt{-c}}{log(-c)} $ for every surface $Sigma$ with Euler characteristic $cleq -2$. We then give more explicit bounds for some surfaces with high Euler characteristic. In particular, we show that the dichromatic numbers of the projective plane $mathbb{N}_1$, the Klein bottle $mathbb{N}_2$, the torus $mathbb{S}_1$, and Dycks surface $mathbb{N}_3$ are all equal to $3$, and that the dichromatic numbers of the $5$-torus $mathbb{S}_5$ and the $10$-cross surface $mathbb{N}_{10}$ are equal to $4$. We also consider the complexity of deciding whether a given digraph or oriented graph embedabble in a fixed surface is $k$-dicolourable. In particular, we show that for any surface, deciding whether a digraph embeddable on this surface is $2$-dicolourable is NP-complete, and that deciding whether a planar oriented graph is $2$-dicolourable is NP-complete unless all planar oriented graphs are $2$-dicolourable (which was conjectured by Neumann-Lara).
Answering a question of Clark and Ehrenborg (2010), we determine asymptotics for the number of permutations of size n that admit the most common excedance set. In fact, we provide a more general bivariate asymptotic using the multivariate asymptotic methods of R. Pemantle and M. C. Wilson. We also consider two applications of our main result. First, we determine asymptotics on the number of permutations of size n which simultaneously avoid the generalized patterns 21-34 and 34-21. Second, we determine asymptotics on the number of n-cycles that admit no stretching pairs.
In a generalized Turan problem, two graphs $H$ and $F$ are given and the question is the maximum number of copies of $H$ in an $F$-free graph of order $n$. In this paper, we study the number of double stars $S_{k,l}$ in triangle-free graphs. We also study an opposite version of this question: what is the maximum number edges/triangles in graphs with double star type restrictions, which leads us to study two questions related to the extremal number of triangles or edges in graphs with degree-sum constraints over adjacent or non-adjacent vertices.
We prove that every graph with $n$ vertices and at least $5n-8$ edges contains the Petersen graph as a minor, and this bound is best possible. Moreover we characterise all Petersen-minor-free graphs with at least $5n-11$ edges. It follows that every graph containing no Petersen minor is 9-colourable and has vertex arboricity at most 5. These results are also best possible.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
