Related papers: R(3,10) <= 41
We consider a set of equations of the form $p_j (x,y) = (10 x+m_j)(10 y + n_j),\,\,x\geq 0, y\geq0$, $j=1,2,3$, such that $\{m_1=7, n_1=3\}$, $\{m_2=n_2=9\}$ and $\{m_3=n_3=1\}$, respectively. It is shown that if $(a(p_j),b(p_j)) \in N…
We improve the $L^{p}\rightarrow L^p$ restriction estimate in $\mathbb{R}^3$ to the range $p>3+3/14$, based on some Kakeya type incidence estimates and the refined decoupling theorem.
A well known conjecture of Burr and Erdos asserts that the Ramsey number $r(Q_n)$ of the hypercube $Q_n$ on $2^n$ vertices is of the order $O(2^n)$. In this paper, we show that $r(Q_n)=O(2^{2n-c n})$ for a universal constant $c>0$,…
The Ramsey number $R(G_1,\dots,G_k)$ is the smallest $n$ such that every $k$-coloring of the edges of $K_n$ contains a monochromatic copy of $G_i$ in color $i$. Ramsey numbers are challenging to compute, and few are known exactly. We use…
Let $Q$ be a finite set of points in the plane. For any set $P$ of points in the plane, $S_{Q}(P)$ denotes the number of similar copies of $Q$ contained in $P$. For a fixed $n$, Erd\H{o}s and Purdy asked to determine the maximum possible…
In this paper, we determine the $r$-colour size Ramsey number of the path $P_k$, up to constants. In particular, for every fixed $r \geq 2$ and $k \geq 100\log r$, we have \[ \widehat{R}_r(P_k)=\Theta((r^2 \log r) \, k).\] Perhaps…
We study the Dyson rank function $N(r,3;n)$, the number of partitions of $n$ with rank $\equiv r \pmod 3$. We investigate the convexity of these functions. We extend $N(r,3;n)$ multiplicatively to the set of partitions, and we determine the…
We estimate the Ramsey number r(T) = r(T,T) for various trees T, obtaining a precise value for r(T) for a large number of trees of diameter 3. Furthermore we prove that all trees of diameter 3 are Ramsey unsaturated as defined by Balister,…
Given a hypergraph $F$ and a number of colours $r$, there exists a hypergraph $H$ of the same girth satisfying $H\longrightarrow (F)_r$. Moreover, for every linear hypergraph $F$ there exists a Ramsey hypergraph $H$ that locally looks like…
We improve the Bochner-Riesz conjecture in $\mathbb{R}^3$ to $\max\{p,p/(p-1)\}\geq3.25$.
Given two graphs $G_1$ and $G_2$, the Ramsey number $r(G_1,G_2)$ refers to the smallest positive integer $N$ such that any graph $G$ with $N$ vertices contains $G_1$ as a subgraph, or the complement of $G$ contains $G_2$ as a subgraph. A…
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 a previous paper with the same title, we gave an upper bound for the exponent of uniform rational approximation to a quadruple of $\mathbb{Q}$-linearly independent real numbers in geometric progression. Here, we explain why this upper…
The square $G^2$ of a graph $G$ is the graph on $V(G)$ with a pair of vertices $uv$ an edge whenever $u$ and $v$ have distance $1$ or $2$ in $G$. Given graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum $N$ such that whenever the…
Let $G$ be a graph and $\Gamma$ a finite abelian group. The zero-sum Ramsey number of $G$ over $\Gamma$, denoted by $R(G, \Gamma)$, is the smallest positive integer $t$ (if it exists) such that any edge-colouring $c:E(K_t)\to\Gamma$…
Recent work in hypergraph Ramsey theory has involved the introduction of a "lifting map" that associates a certain $3$-uniform hypergraph to a given graph, bounding cliques in a predictable way. In this paper, we interpret the lifting map…
Gy\'{a}rf\'{a}s et al. determined the asymptotic value of the diagonal Ramsey number of $\mathcal{C}^k_n$, $R(\mathcal{C}^k_n,\mathcal{C}^k_n),$ generating the same result for $k=3$ due to Haxell et al. Recently, the exact values of the…
We show that the size-Ramsey number of the $\sqrt{n} \times \sqrt{n}$ grid graph is $O(n^{5/4})$, improving a previous bound of $n^{3/2 + o(1)}$ by Clemens, Miralaei, Reding, Schacht, and Taraz.
It is known that $|\zeta(1+ it)|\ll (\log t)^{2/3}$. This paper provides a new explicit estimate, viz.\ $|\zeta(1+ it)|\leq 3/4 \log t$, for $t\geq 3$. This gives the best upper bound on $|\zeta(1+ it)|$ for $t\leq 10^{2\cdot 10^{5}}$.
We show that any graded digraph $D$ on $n$ vertices with maximum degree $\Delta$ has an oriented Ramsey number of at most $C^\Delta n$ for some absolute constant $C > 1$, improving upon a recent result of Fox, He, and Wigderson. In…