Related papers: A New Lower Bound for van der Waerden Numbers

Let $p>3$ be a prime. In this paper, we obtain the congruences for $$\sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^3}{(-8)^k},\ \sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{3k}k}{(-192)^k},\…

We prove a couple of related theorems including Legendre's and Andrica's conjecture. Key to the proofs is an algorithm that delivers the exact upper bound on the greatest gap that can occur in a combinatorial game with the set of P primes…

Let $p$ be a prime, and let $f(x)$ be an integer-valued polynomial. By a combinatorial approach, we obtain a nontrivial lower bound of the $p$-adic order of the sum $$\sum_{k=r(mod p^{\beta})}\binom{n}{k}(-1)^k f([(k-r)/p^{\alpha}]),$$…

Let $r(k)$ denote the maximum number of edges in a $k$-uniform intersecting family with covering number $k$. Erd\H{o}s and Lov\'asz proved that $ \lfloor k! (e-1) \rfloor \leq r(k) \leq k^k.$ Frankl, Ota, and Tokushige improved the lower…

In 1862 Wolstenholme proved that for any prime $p\ge 5$ the numerator of the fraction $$ 1+\frac 12 +\frac 13+...+\frac{1}{p-1} $$ written in reduced form is divisible by $p^2$, $(2)$ and the numerator of the fraction $$ 1+\frac{1}{2^2}…

For integers $k,r\geq 2$, the diagonal Ramsey number $R_r(k)$ is the minimum $N\in\mathbb{N}$ such that every $r$-coloring of the edges of a complete graph on $N$ vertices yields on a monochromatic subgraph on $k$ vertices. Here we make a…

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…

Let $P$ be a convex $d$-polytope and $0 \leq k \leq d-1$. In 2023, this author proved the following inequalities, resolving a question of B\'ar\'any: \[ \frac{f_k(P)}{f_0(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose k} +…

Given two graphs $G$ and $H$, the {\it rainbow number} $rb(G,H)$ for $H$ with respect to $G$ is defined as the minimum number $k$ such that any $k$-edge-coloring of $G$ contains a rainbow $H$, i.e., a copy of $H$, all of whose edges have…

For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is guaranteed…

For every $p\leq n$ positive integer we obtain the lower bound $(3-\frac{1}{p+1})n^2-\big(2\binom{2p}{p+1}-\binom{2p-2}{p-1}+2\big)n$ for the rank of the $n\times n$ matrix multiplication. This bound improves the previous one…

In 1927, Artin conjectured that any integer other than -1 or a perfect square generates the multiplicative group $\mathbb{Z}/p\mathbb{Z}^\times$ for infinitely many $p$. In \cite{MoSt}, Moree and Stevenhagen considered a two-variable…

In this paper, we prove some supercongruences via the Wilf-Zeilberger method. For instance, for any odd prime $p$ and positive integer $r$ and $\delta\in\{1,2\}$, we have \begin{align*} \sum_{n=0}^{(p^r-1)/\delta}…

Let $p$ be an odd prime such that $p \equiv 3\;{\rm mod}\;4$ and $n$ be an odd integer. In this paper, two new families of $p$-ary sequences of period $N = \frac{p^n-1}{2}$ are constructed by two decimated $p$-ary m-sequences $m(2t)$ and…

The Raney numbers $R_{p,r}(k)$ are a two-parameter generalization of the Catalan numbers. In this paper, we obtain a recurrence relation for the Raney numbers which is a generalization of the recurrence relation for the Catalan numbers.…

Let $k$ be an integer. Two vertex $k$-colorings of a graph are \emph{adjacent} if they differ on exactly one vertex. A graph is \emph{$k$-mixing} if any proper $k$-coloring can be transformed into any other through a sequence of adjacent…

Let $p$ be a prime and ${\mathfrak P}_p$ the set of positive integers which are prime to $p$. Recently, Wang and Cai proved that for every positive integer $r$ and prime $p>2$ $$ \sum_{\substack{i+j+k=p^r\\ i,j,k\in{\mathfrak P}_p}}…

Let $p$ be an odd prime and let $x$ be a $p$-adic integer. In this paper, we establish supercongruences for $$ \sum_{k=0}^{p-1}\frac{\binom{x}{k}\binom{x+k}{k}(-4)^k}{(dk+1)\binom{2k}{k}}\pmod{p^2} $$ and $$…

A regularity lemma for polynomials provides a decomposition in terms of a bounded number of approximately independent polynomials. Such regularity lemmas play an important role in numerous results, yet suffer from the familiar shortcoming…

We investigate when the sequence of binomial coefficients \binom{k}{i} modulo a prime p, for a fixed positive integer k, satisfies a linear recurrence relation of (positive) degree h in the finite range 0\le i\le k. In particular, we prove…