Minimal degree univariate piecewise polynomials with prescribed Sobolev regularity

For $k\in \{1,2,3,\dots \}$, we construct an even compactly supported piecewise polynomial ${\psi }_k$ whose Fourier transform satisfies $A_k{(1+{\omega }^2)}^{?k}\le {\stackrel{?}{\psi }}_k\left(\omega \right)\le B_k{(1+{\omega }^2)}^{?k}$, $\omega \in \mathbb{R}$, for some constants $B_k\ge A_k>0$. The degree of ${\psi }_k$ is shown to be minimal, and is strictly less than that of Wendland’s function ${?}_{1,k?1}$ when $k>2$. This shows that, for $k>2$, Wendland’s piecewise polynomial ${?}_{1,k?1}$ is not of minimal degree if one places no restrictions on the number of pieces.     

Ramsey numbers of odd cycles versus larger even wheels

The generalized Ramsey number $R(G_1,G_2)$ is the smallest positive integer $N$ such that any red–blue coloring of the edges of the complete graph $K_N$ either contains a red copy of $G_1$ or a blue copy of $G_2$. Let $C_m$ denote a cycle of length $m$ and $W_n$ denote a wheel with $n+1$ vertices. In 2014, Zhang, Zhang and Chen determined many of the Ramsey numbers $R(C_{2k+1},W_n)$ of odd cycles versus larger wheels, leaving open the particular case where $n=2j$ is even and $k<j<3k∕2$. They conjectured that for these values of $j$ and $k$, $R(C_{2k+1},W_{2j})=4j+1$. In 2015, Sanhueza-Matamala confirmed this conjecture asymptotically, showing that $R(C_{2k+1},W_{2j})\le 4j+334$. In this paper, we prove the conjecture of Zhang, Zhang and Chen for almost all of the remaining cases. In particular, we prove that $R(C_{2k+1},W_{2j})=4j+1$ if $j?k\ge 251$, $k<j<3k∕2$, and $j\ge 212299$.     

Partitions of natural numbers with the same weighted representation functions

TextFor any given two positive integers $k_1$ and $k_2$, and any set A of nonnegative integers, let $r_{k_1,k_2}(A,n)$ denote the number of solutions of the equation $n=k_1{a}_1+k_2{a}_2$ with $a_1,a_2\in A$. In this paper, we determine all pairs $k_1,k_2$ of positive integers for which there exists a set $A?\mathbb{N}$ such that $r_{k_1,k_2}(A,n)=r_{k_1,k_2}(\mathbb{N}?A,n)$ for all $n?n_0$. We also pose several problems for further research.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EnezEsJl0OY.     

Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

For bipartite graphs $G_1,G_2,\dots ,G_k$, the bipartite Ramsey number $b(G_1,G_2,\dots ,G_k)$ is the least positive integer $b$ so that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a copy of $G_i$ in the $i$th color for some $i$. In this paper, our main focus will be to bound the following numbers: $b(C_{2t_1},C_{2t_2})$ and $b(C_{2t_1},C_{2t_2},C_{2t_3})$ for all $t_i\ge 3,$$b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4})$ for $3\le t_i\le 9,$ and $b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4},C_{2t_5})$ for $3\le t_i\le 5.$ Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result.     

Sums of powers of Catalan triangle numbers

In this paper, we consider combinatorial numbers ${\left(C_{m,k}\right)}_{m\ge 1,k\ge 0}$, mentioned as Catalan triangle numbers where $C_{m,k}?\left(\genfrac{}{}{0ex}{}{m?1}{k}\right)?\left(\genfrac{}{}{0ex}{}{m?1}{k?1}\right)$. These numbers unify the entries of the Catalan triangles $B_{n,k}$ and $A_{n,k}$ for appropriate values of parameters $m$ and $k$, i.e., $B_{n,k}=C_{2n,n?k}$ and $A_{n,k}=C_{2n+1,n+1?k}$. In fact, these numbers are suitable rearrangements of the known ballot numbers and some of these numbers are the well-known Catalan numbers $C_n$ that is $C_{2n,n?1}=C_{2n+1,n}=C_n$.We present identities for sums (and alternating sums) of $C_{m,k}$, squares and cubes of $C_{m,k}$ and, consequently, for $B_{n,k}$ and $A_{n,k}$. In particular, one of these identities solves an open problem posed in Gutiérrez et al. (2008). We also give some identities between ${\left(C_{m,k}\right)}_{m\ge 1,k\ge 0}$ and harmonic numbers ${\left(H_n\right)}_{n\ge 1}$. Finally, in the last section, new open problems and identities involving ${\left(C_n\right)}_{n\ge 0}$ are conjectured.     

Minimal homogeneous Steiner 2-(v,3) trades

A Steiner 2-$(v,3)$ trade is a pair $(T_1,T_2)$ of disjoint partial Steiner triple systems, each on the same set of $v$ points, such that each pair of points occurs in $T_1$ if and only if it occurs in $T_2$. A Steiner 2-$(v,3)$ trade is called d-homogeneous if each point occurs in exactly d blocks of $T_1$ (or $T_2$). In this paper we construct minimal d-homogeneous Steiner 2-$(v,3)$ trades of foundation $v$ and volume $\mathit{dv}/3$ for sufficiently large values of $v$. (Specifically, $v>3(1.75d^2+3)$ if $v$ is divisible by 3 and $v>d(4^{d/3+1}+1)$ otherwise.)     

Proper connection and size of graphs

An edge-coloured graph $G$ is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a connected graph $G,$ denoted by $\mathrm{pc}\left(G\right)$, is the smallest number of colours that are needed in order to make $G$ properly connected. Our main result is the following: Let $G$ be a connected graph of order $n$ and $k\ge 2$. If $\left|E\left(G\right)\right|\ge \left(\genfrac{}{}{0ex}{}{n?k?1}{2}\right)+k+2$, then $\mathrm{pc}\left(G\right)\le k$ except when $k=2$ and $G\in \{G_1,G_2\},$ where $G_1=K_1\vee (2K_1+K_2)$ and $G_2=K_1\vee (K_1+2K_2).$     

On a problem of partitions of the set of nonnegative integers with the same representation functions

Dombi has shown that the set $\mathbb{N}$ of all non-negative integers can be partitioned into two subsets with identical representation functions. In this paper, we prove that one cannot partition $\mathbb{N}$ into more than two subsets with identical representation functions, while for any integer $k\ge 3$ there is a partition $\mathbb{N}=A_1\cup ?\cup A_k$ such that $A_i$ and $A_{k+1?i}$ have the same representation function for any integer $1\le i\le k$.     

A new forbidden pair for 6-contractible edges

An edge of a $k$-connected graph is said to be $k$-contractible if the contraction of the edge results in a $k$-connected graph. If every $k$-connected graph with no $k$-contractible edge has either $H_1$ or $H_2$ as a subgraph, then an unordered pair of graphs $\{H_1,H_2\}$ is said to be a forbidden pair for $k$-contractible edges. We prove that $\{K_1+3K_2,K_1+(P_3\cup K_2)\}$ is a forbidden pair for 6-contractible edges, which is an extension of a previous result due to Ando and Kawarabayashi.