Related papers: New Bounds on van der Waerden-type Numbers for Gen…

In this paper arithmetic progressions on the integers and the integers modulo n are extended to graphs. This allows for the definition of the anti-van der Waerden number of a graph. Much of the focus of this paper is on 3-term arithmetic…

For every even positive integer $k\ge 4$ let $f(n,k)$ denote the minimim number of colors required to color the edges of the $n$-dimensional cube $Q_n$, so that the edges of every copy of $k$-cycle $C_k$ receive $k$ distinct colors.…

For two relatively prime square-free positive integers $a$ and $b$, we study integers of the form $a p+b P_{2}$ and give a new lower bound for the number of such representations, where $a p$ and $b P_{2}$ are both square-free, $p$ denote a…

Let $d>r\ge 0$ be integers. For positive integers $a,b,c$, if any term of the arithmetic progression $\{r+dn:\ n=0,1,2,\ldots\}$ can be written as $ax^2+by^2+cz^2$ with $x,y,z\in\mathbb{Z}$, then the form $ax^2+by^2+cz^2$ is called…

A tri-colored sum-free set in an abelian group $H$ is a collection of ordered triples in $H^3$, $\{(a_i,b_i,c_i)\}_{i=1}^m$, such that the equation $a_i+b_j+c_k=0$ holds if and only if $i=j=k$. Using a variant of the lemma introduced by…

Let $A(n,d)$ (respectively $A(n,d,w)$) be the maximum possible number of codewords in a binary code (respectively binary constant-weight $w$ code) of length $n$ and minimum Hamming distance at least $d$. By adding new linear constraints to…

An abstract, Hales-Jewett type extension of the polynomial van der Waerden Theorem [J. Amer. Math. Soc. 9 (1996),725-753] is established: Theorem. Let r,d,q \in \N. There exists N \in \N such that for any r-coloring of the set of subsets of…

We present a new positive lower bound for the minimum value taken by a polynomial P with integer coefficients in k variables over the standard simplex of R^k, assuming that P is positive on the simplex. This bound depends only on the number…

We created and parallelized two SAT solvers to find new bounds on some Ramsey-type numbers. For $c > 0$, let $R_c(L)$ be the least $n$ such that for all $c$-colorings of the $[n]\times [n]$ lattice grid there will exist a monochromatic…

For a given length and a given degree and an arbitrary partition of the positive integers, there always is a cell containing a polynomial progression of that length and that degree; moreover, the coefficients of the generating polynomial…

Let $AP_k=\{a,a+d,\ldots,a+(k-1)d\}$ be an arithmetic progression. For $\epsilon>0$ we call a set $AP_k(\epsilon)=\{x_0,\ldots,x_{k-1}\}$ an $\epsilon$-approximate arithmetic progression if for some $a$ and $d$, $|x_i-(a+id)|<\epsilon d$…

For nonnegative integers $q,n,d$, let $A_q(n,d)$ denote the maximum cardinality of a code of length $n$ over an alphabet $[q]$ with $q$ letters and with minimum distance at least $d$. We consider the following upper bound on $A_q(n,d)$. For…

Let $\operatorname{r_{max}}(n,d)$ be the maximum Waring rank for the set of all homogeneous polynomials of degree $d>0$ in $n$ indeterminates with coefficients in an algebraically closed field of characteristic zero. To our knowledge, when…

For two coprime positive integers $a,b$, let $T(a,b)=\{ ax+by : x,y\in \mathbb{Z}_{\ge 0} \} $ and let $s(a,b)=ab-a-b$. It is well known that all integers which are greater than $s(a,b)$ are in $T(a,b)$. Let $\pi (a, b)$ be the number of…

Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: R(3,10) <= 42, R(3,11) <= 50, R(3,13) <= 68, R(3,14) <= 77, R(3,15) <= 87, and R(3,16) <= 98. All of them are improvements by one over…

The Tur\'{a}n number $T(n,\alpha+1,r)$ is the minimum number of edges in an $n$-vertex $r$-graph whose independence number does not exceed $\alpha$. For each $r\geq 2$, there exists $t_*(r)$ such that $T(n,\alpha+1,r) = t_*(r) \: n^r \:…

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order…

Consider the divisor sum $\sum_{n\leq N}\tau(n^2+2bn+c)$ for integers $b$ and $c$ which satisfy certain extra conditions. For this average sum we obtain an explicit upper bound, which is close to the optimal. As an application we improve…

Let $G=(V,E)$ be a simple graph with $n$ vertices and $m$ edges, $P(G,k)$ be the chromatic polynomial of $G$, and $P(G,L)$ be the number of $L$-colorings of $G$ for any $k$-assignment $L$. In this article, we show that when $k\ge m-1\ge 3$,…

A Hardy-Littlewood triple is a 3-tuple of integers with the form $(n, n+2, n+6)$. In this paper, we study Hardy-Littlewood triples of the form $(p, P_{a}, P_{b})$ and improve the upper and lower bound orders of it, where $p$ is a prime and…