We introduce two Ramsey classes of finite relational structures. The first class contains finite structures of the form $\left( A,\,\left( {{I}_{i}} \right)_{i=1}^{n},\le ,\left( {{\underline{\prec }}_{i}} \right)_{i=1}^{n} \right)$, where $\le$ is a total ordering on $A$ and ${{\underline{\prec }}_{i}}$ is a linear ordering on the set $\left\{ a\,\in \,A\,:\,{{I}_{i}}\left( a \right) \right\}$. The second class contains structures of the form a $\left( a,\le ,\left( {{i}_{i}} \right)_{i=1}^{n},\underline{\prec } \right)$, where $\left( A,\,\le \right)$ is a weak ordering and $\underline{\prec }$ is a linear ordering on $A$ such that $A$ is partitioned by $\left\{ a\,\in \,A\,:\,{{I}_{i}}\left( a \right) \right\}$ into maximal chains in the partial ordering $\le$ and each $\left\{ a\,\in \,A\,:\,{{I}_{i}}\left( a \right) \right\}$ is an interval with respect to $\underline{\prec }$.

A well-known theorem of Schütte (1963) gives a sharp lower bound for the ratio of the maximum and minimum distances between $n\,+\,2$ points in $n$-dimensional Euclidean space. In this note we adapt Bárány’s elegant proof (1994) of this theorem to the space $\ell _{4}^{n}$. This gives a new proof that the largest cardinality of an equilateral set in $\ell _{4}^{n}$ is $n\,+\,1$ and gives a constructive bound for an interval $\left( 4\,-\,{{\varepsilon }_{n}},\,4\,+\,{{\varepsilon }_{n}} \right)$ of values of $p$ close to 4 for which it is known that the largest cardinality of an equilateral set in $\ell _{p}^{n}$ is $n\,+\,1$.

Let $G$ be a finite group and let $\mathcal{F}$ be a class of groups. Then ${{Z}_{\mathcal{F}\Phi }}\left( G \right)$ is the $\mathcal{F}\Phi$-hypercentre of $G$, which is the product of all normal subgroups of $G$ whose non-Frattini $G$-chief factors are $\mathcal{F}$-central in $G$. A subgroup $H$ is called $\mathcal{M}$-supplemented in a finite group $G$ if there exists a subgroup $B$ of $G$ such that $G\,=\,HB\,\text{and}\,{{H}_{1}}B$ is a proper subgroup of $G$ for any maximal subgroup ${{H}_{1}}$ of $H$. The main purpose of this paper is to prove the following: Let $E$ be a normal subgroup of a group $G$. Suppose that every noncyclic Sylow subgroup $P\,\text{of}\,{{F}^{*}}\left( E \right)$ has a subgroup $D$ such that $1\,<\,\left| D \right|\,<\left| P \right|$ and every subgroup $H\,\text{of}\,P$ with order $\left| H \right|\,=\,\left| D \right|$ is $\mathcal{M}$-supplemented in $G$, then $E\,\le \,{{Z}_{\mathcal{U}\Phi }}\left( G \right)$.

Let $\mathcal{A}$ be a line arrangement in the complex projective plane ${{\mathbb{P}}^{2}}$ and let $M$ be its complement. A rank one local system $\mathcal{L}$ on $M$ is admissible if, roughly speaking, the cohomology groups ${{H}^{m}}\left( M,\,\mathcal{L} \right)$ can be computed directly from the cohomology algebra ${{H}^{*}}\left( M,\,\mathbb{C} \right)$. In this work, we give a sufficient condition for the admissibility of all rank one local systems on $M$. As a result, we obtain some properties of the characteristic variety ${{\mathcal{V}}_{1}}\left( M \right)$ and the Resonance variety ${{\mathcal{R}}_{1}}\left( M \right)$.