For graphs $G$, $F_1$, $F_2$, we write $G\to (F_1,F_2)$ if for every red-blue colouring of the edge set of $G$ we have a red copy of $F_1$ or a blue copy of $F_2$ in $G$. The size Ramsey number $\widehat{r}(F_1,F_2)$ is the minimum number of edges of a graph $G$ such that $G\to (F_1,F_2)$. Erdős and Faudree proved that for the cycle $C_n$ of length $n$ and for $t\ge 2$ matchings $t{K}_2$, the size Ramsey number $\widehat{r}(t{K}_2,C_n)<n+(4t+3)\sqrt{n}$. We improve their upper bound for $t=2$ and $t=3$ by showing that $\widehat{r}(2K_2,C_n)\le n+2\sqrt{3n}+9$ for $n\ge 12$ and $\widehat{r}(3K_2,C_n)<n+6\sqrt{n}+9$ for $n\ge 25$.

Let $G$ be a finite group. An element $g\in G$ is called a vanishing element if there exists an irreducible complex character $\chi$ of $G$ such that $\chi \left(g\right)=0$. Denote by $\mathrm{Vo}\left(G\right)$ the set of orders of vanishing elements of $G$. Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let $G$ be a finite group and $M$ a finite nonabelian simple group such that $\mathrm{Vo}\left(G\right)=\mathrm{Vo}\left(M\right)$ and $\left|G\right|=\left|M\right|$. Then $G\cong M$. We answer in affirmative this conjecture for $M=Sz\left(q\right)$, where $q=2^{2n+1}$ and either $q-1$, $q-\sqrt{2q}+1$ or $q+\sqrt{2q}+1$ is a prime number, and $M=F_4\left(q\right)$, where...

Let $d$ be a square free integer and $L_d:=\mathbb{Q}({\zeta }_8,\sqrt{d})$. In the present work we determine all the fields $L_d$ such that the $2$-class group, ${\mathrm{Cl}}_2\left(L_d\right)$, of $L_d$ is of type $(2,4)$ or $(2,2,2)$.

CONTENTSIntroduction.......................................................................................61. Definition of certain special forms...........................................62. Statement of results...................................................................83. Proof of Theorem 2.....................................................................94. Preliminaries for Theorem 1.....................................................105. Further preliminaries for Theorem...

A nonincreasing sequence $\pi =(d_1,...,d_n)$ of nonnegative integers is a graphic sequence if it is realizable by a simple graph $G$ on $n$ vertices. In this case, $G$ is referred to as a realization of $\pi$. Given two graphs $G_1$ and $G_2$, A. Busch et al. (2014) introduced the potential-Ramsey number of $G_1$ and $G_2$, denoted by $r_{\mathrm{pot}}(G_1,G_2)$, as the smallest nonnegative integer $m$ such that for every $m$-term graphic sequence $\pi$, there is a realization $G$ of $\pi$ with $G_1\subseteq G$ or with $G_2\subseteq \overline{G}$, where $\overline{G}$ is the complement of $G$. For $t\ge 2$ and $0\le k\le \lfloor \frac{t}{2}\rfloor$, let $K_t^{-k}$ be the graph...

For an algebraic number field $K$ with $3$-class group ${\mathrm{Cl}}_3\left(K\right)$ of type $(3,3)$, the structure of the $3$-class groups ${\mathrm{Cl}}_3\left(N_i\right)$ of the four unramified cyclic cubic extension fields $N_i$, $1\le i\le 4$, of $K$ is calculated with the aid of presentations for the metabelian Galois group ${\mathrm{G}}_3^2\left(K\right)=\mathrm{Gal}\left({\mathrm{F}}_3^2\left(K\right)\right|K)$ of the second Hilbert $3$-class field ${\mathrm{F}}_3^2\left(K\right)$ of $K$. In the case of a quadratic base field $K=\mathbb{Q}\left(\sqrt{D}\right)$ it is shown that the structure of the $3$-class groups of the four $S_3$-fields $N_1,...,N_4$ frequently determines the type of principalization of the $3$-class group of $K$ in $N_1,...,N_4$. This...

We give the characterization of the unit group of ${𝔽}_{q}SL(2,{\mathbb{Z}}_5)$, where ${𝔽}_q$ is a finite field with $q=p^k$ elements for prime $p>5,$ and $SL(2,{\mathbb{Z}}_5)$ denotes the special linear group of $2\times 2$ matrices having determinant $1$ over the cyclic group ${\mathbb{Z}}_5$.

We study sums and products in a field. Let $F$ be a field with $\mathrm{ch}\left(F\right)\ne 2$, where $\mathrm{ch}\left(F\right)$ is the characteristic of $F$. For any integer $k\ge 4$, we show that any $x\in F$ can be written as $a_1+\cdots +a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1\cdots a_k=1$, and that for any $\alpha \in F\setminus \left\{0\right\}$ we can write every $x\in F$ as $a_1\cdots a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1+\cdots +a_k=\alpha$. We also prove that for any $x\in F$ and $k\in \{2,3,\cdots \}$ there are $a_1,\cdots ,a_{2k}\in F$ such that $a_1+\cdots +a_{2k}=x=a_1\cdots a_{2k}$.

We consider the equation $$-y^{\text{'}}\left(x\right)+q\left(x\right)y(x-\varphi \left(x\right))=f\left(x\right),x\in ℝ,$$ where $\varphi$ and $q$ ($q\ge 1$) are positive continuous functions for all $x\in ℝ$ and $f\in C\left(ℝ\right)$. By a solution of the equation we mean any function $y$, continuously differentiable everywhere in $ℝ$, which satisfies the equation for all $x\in ℝ$. We show that under certain additional conditions on the functions $\varphi$ and $q$, the above equation has a unique solution $y$, satisfying the inequality $$\parallel y^{\text{'}}{\parallel }_{C\left(ℝ\right)}+{\parallel qy\parallel }_{C\left(ℝ\right)}\le c{\parallel f\parallel }_{C\left(ℝ\right)},$$ where the constant $c\in (0,\infty )$ does not depend on the choice of $f$.