Abstract We consider the simplices $$\begin{aligned} K_n^A=\{x\in {\mathbb {R}}^{n+1}:x_1\ge x_2\ge \cdots \ge x_{n+1},x_1-x_{n+1}\le 1,\,x_1+\cdots +x_{n+1}=0\} \end{aligned}$$ K n A = {</m...

A $k$-uniform tight cycle is a hypergraph with cyclic ordering of its vertices such that edges are all the sets size $k$ formed by consecutive in ordering.We prove every red-blue edge-coloured $K_n^{(4)}$ contains red and blue vertex-disjoint together cover $n-o(n)$ vertices. Moreover, we $K_n^{(5)}$ four monochromatic cycles

A factorisation x = u 1 2 ? $x u_1 u_2 \cdots$ of an infinite word $x$ on alphabet X $X$ is called ‘monochromatic’, for a given colouring the finite words ? $X^*$ , if each i $u_i$ same colour. Wojcik and Zamboni proved that periodic only every there monochromatic . On other hand, it follows from Ramsey's theorem that, any suffix having factorisation. ‘super-monochromatic’ k n $u_{k_1} u_{k_2} ...

The sum of distances between all the pairs of vertices in a connected graph is known as the {it Wiener index} of the graph. In this paper, we obtain the Wiener index of edge complements of stars, complete subgraphs and cycles in $K_n$.

Ramsey proved that for every positive integer $n$, sufficiently large graph contains an induced $K_n$ or $\overline{K}_n$. Among the many extensions of Ramsey's Theorem there is analogue connected graphs: $K_n$, $K_{1,n}$, $P_n$. In this paper, we establish 2-connected graphs. particular, prove exceeding two, one following as subgraph: a subdivision $K_{2,n}$, $K_{2,n}$ with edge between two ve...

Given $d\in \mathbb{N}$, $g\in \mathbb{N} \cup\{0\}$, and an integral vector $\kappa=(k_1,\dots,k_n)$ such that $k_i>-d$ $k_1+\dots+k_n=d(2g-2)$, let $\Omega^d\mathcal{M}_{g,n}(\kappa)$ denote the moduli space of meromorphic $d$-differentials on Riemann surfaces genus $g$ whose zeros poles have orders prescribed by $\kappa$. We show carries a canonical volume form is parallel with respect to it...

In this article we discuss classic theorems from Convex Geometry in the context of topological drawings and beyond. a simple drawing complete graph $$K_n$$ , any two edges share at most one point: either common vertex or point where they cross. Triangles can be viewed as convex sets. This gives link to geometry. As our main result, present generalization Kirchberger’s theorem that is purely com...