Minimal Filling $K$-Systems of Curves
In this paper, we determine the exact minimal number of curves in a filling $k$-system on an oriented surface of genus $g$ for any positive integers $k$ and $g$.
math.GT↗
arXiv subjects
Wujie Shen
Publications and source records attributed to Wujie Shen.
An exponential improvement for Ramsey lower bounds
We prove a new lower bound on the Ramsey number $r(\ell, C\ell)$ for any constant $C > 1$ and sufficiently large $\ell$, showing that there exists $\varepsilon=\varepsilon(C)> 0$ such that \[ r(\ell, C\ell) \geq \left(p_C^{-1/2} + \varepsilon\right)^\ell, \] where $p_C \in (0, 1/2)$ is the unique solution to $C = \frac{\log p_C}{\log(1 - p_C)}$. This provides the first exponential improvement over the classical lower bound obtained by Erdős in 1947.
math.CO↗
$\partial$-invariant path generators for digraphs
We study the structure of the space $Ω_3(G)$ of $\partial$-invariant 3-paths in a directed graph $G$. We prove that $Ω_3(G)$ admits a basis consisting of trapezohedral paths $τ_m$ ($m \ge 2$) and their merging images. Moreover, we provide an explicit construction of such a basis and, as a consequence, obtain an algorithm with time complexity $O(|V(G)|^5)$ for computing the dimension and a basis of $Ω_3(G)$ for any finite digraph.
math.CO↗
A Linear Bound on the Diameter of the Kakimizu Complex for Hyperbolic Knots
This paper focuses on the Kakimizu complex of a hyperbolic knot $K$. We define a complex $IS_\ell(K)$ to study incompressible Seifert surfaces of genus at most $\ell$, and prove that it is connected and that its diameter admits a linear upper bound in terms of $\ell$. As a corollary, we show that the diameter of the Kakimizu complex $MS(K)$ of a hyperbolic knot grows linearly with the genus $g$, confirming a conjecture of Sakuma--Shackleton. More precisely, it is bounded above by $6g-4$.
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Closed geodesics on hyperbolic surfaces with few intersections
We prove that, if a closed geodesic $Γ$ on a complete finite type hyperbolic surface has at least 2 self-intersections, then the length of $Γ$ has an lower bound $2\log(5+2\sqrt6)$, and the lower bound is sharp, attained on a corkscrew geodesic on a thrice punctured sphere.
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Nonsimple closed geodesics with given intersection number on hyperbolic surfaces
We prove that the minimal length of a closed geodesic with self-intersection number $k$ on any finite-type hyperbolic surface is $2\cosh^{-1}(1+2k)$ for $k>1750$. This improves the previously known threshold $k > 10^{13350}$. Our proof is independent of it.
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Uniqueness of Maximal Curve Systems on Punctured Projective Planes
We show that the maximal $1$-system of loops in a punctured projective plane is unique up to the mapping class group action if and only if the number of punctures is at most five.
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Minimal length of nonsimple closed geodesics on hyperbolic surfaces
In the present paper, we show that the minimal length of closed geodesics on finite-type hyperbolic surfaces with self-intersection number $k$ has order $2\log k$ as $k$ gets large.
math.GT↗