Variants of the Erdős–Szekeres and Erdős–Hajnal Ramsey problems

Given integers $\ell ,n$, the $\ell$th power of the path $P_n$ is the ordered graph $P_n^{\ell }$ with vertex set $v_1<v_2<\cdots <v_n$ and all edges of the form $v_i{v}_j$ where $|i-j|\le \ell$. The Ramsey number $r(P_n^{\ell },P_n^{\ell })$ is the minimum $N$ such that every 2-coloring of $\left(\genfrac{}{}{0ex}{}{\left[N\right]}{2}\right)$ results in a monochromatic copy of $P_n^{\ell }$. It is well-known that $r(P_n^1,P_n^1)={(n-1)}^2+1$. For $\ell >1$, Balko–Cibulka–Král–Kynčl proved that $r(P_n^{\ell },P_n^{\ell })<c_{\ell }{n}^{128\ell }$ and asked for the growth rate for fixed $\ell$. When $\ell =2$, we improve this upper bound substantially by proving $r(P_n^2,P_n^2)<c{n}^{19.5}$. Using this result, we determine the correct tower growth rate of the $k$-uniform hypergraph Ramsey number of a $(k+1)$-clique versus an ordered tight path. Finally, we consider an ordered version of the classical Erdős–Hajnal hypergraph Ramsey problem, improve the tower height given by the trivial upper bound, and conjecture that this tower height is optimal.     

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.     

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.     

Dynamics of epidemics in homogeneous/heterogeneous populations and the spreading of multiple inter-related infectious diseases: Constant-sign periodic solutions for the discrete model

We consider the following model that describes the dynamics of epidemics in homogeneous/heterogeneous populations as well as the spreading of multiple inter-related infectious diseases:$u_i(k)=\sum _{\ell =k-{\tau }_i}^{k-1}{g}_i(k,\ell )f_i(\ell ,u_1(\ell ),u_2(\ell ),\dots ,u_n(\ell )),k\in \mathbf{Z},1⩽i⩽n\text{.}$Our aim is to establish criteria such that the above system has one or multiple constant-sign periodic solutions $(u_1,u_2,\dots ,u_n)$, i.e., for each $1⩽i⩽n$, $u_i$ is periodic and ${\theta }_i{u}_i⩾0$ where ${\theta }_i\in \{1,-1\}$ is fixed. Examples are also included to illustrate the results obtained.     

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.     

On (Kq,k) vertex stable graphs with minimum size

A graph $G$ is a $(K_q,k)$ vertex stable graph if it contains a $K_q$ after deleting any subset of $k$ vertices. We give a characterization of $(K_q,k)$ vertex stable graphs with minimum size for $q=3,4,5$.     

Distinct distances between a collinear set and an arbitrary set of points

We consider the number of distinct distances between two finite sets of points in ${\mathbb{R}}^k$, for any constant dimension $k\ge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O\left(1\right)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then

$\Omega \left(min\left\{n^{2∕3}{m}^{2∕3},\frac{n^{10∕11}{m}^{4∕11}}{{log}^{2∕11}m},n^2,m^2\right\}\right).$

Without the assumption on $P_2$, there exist sets $P_1$, $P_2$ as above, with only $O(m+n)$ distinct distances between them.     

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 $j\ge 212299$.     

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.