相关论文: New Lower Bound Formulas for Multicolored Ramsey N…

Upper bounds on the maximum number of codewords in a binary code of a given length and minimum Hamming distance are considered. New bounds are derived by a combination of linear programming and counting arguments. Some of these bounds…

The Ramsey multiplicity problem asks for the minimum asymptotic density of monochromatic labelled copies of a graph $H$ in a red/blue colouring of the edges of $K_n$. We introduce an off-diagonal generalization in which the goal is to…

Behavior of the entropy numbers of classes of multivariate functions with mixed smoothness is studied here. This problem has a long history and some fundamental problems in the area are still open. The main goal of this paper is to develop…

The Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple…

Given an equation, the integers $[n] = \{1, 2, \dots, n\}$ as inputs, and the colors red and blue, how can we color $[n]$ in order to minimize the number of monochromatic solutions to the equation, and what is the minimum? The answer is…

The Rado number of an equation is a Ramsey-theoretic quantity associated to the equation. Let $\mathcal{E}$ be a linear equation. Denote by $\operatorname{R}_r(\mathcal{E})$ the minimal integer, if it exists, such that any $r$-coloring of…

The double star $S(m_1,m_2)$ is obtained from joining the centres of a star with $m_1$ leaves and a star with $m_2$ leaves. We give a short proof of a new upper bound on the two-colour Ramsey number of $S(m_1,m_2)$, for positive $m_1,m_2$…

We propose two families of asymptotically local minimax lower bounds on parameter estimation performance. The first family of bounds applies to any convex, symmetric loss function that depends solely on the difference between the estimate…

Let $F_n$, $W_n$, and $\widehat{K}_n$ be the graphs obtained by joining a vertex to $n$ independent edges, a cycle and a path of order $n-1$, respectively. In this paper, we give new bounds for the Ramsey numbers $R(F_n,F_m)$ and…

Ramsey theory is a central and active branch of combinatorics. Although Ramsey numbers for graphs have been extensively investigated since Ramsey's work in the 1930s, there is still an exponential gap between the best known lower and upper…

We derive exact and sharp lower bounds for the number of monochromatic generalized Schur triples $(x,y,x+ay)$ whose entries are from the set $\{1,\dots,n\}$, subject to a coloring with two different colors. Previously, only asymptotic…

We obtain a lower bound for a number of primes in tuples. As applications, we obtain a lower bound for the Romanoff type representation functions.

We prove a lower bound on the dimension of the set of maximal border subrank tensors. This is the first such bound of its type.

Upper bounds are given for the weight distribution of binary weakly self-dual codes. To get these new bounds, we introduce a novel method of utilizing unitary operations on Hilbert spaces. This method is motivated by recent progress on…

We give an explicitly computable lower bound for the arithmetic self-intersection number of the dualizing sheaf on a large class of arithmetic surfaces. If some technical conditions are satisfied, then this lower bound is positive. In…

We provide two novel constructions of $r$ edge-disjoint $K_{k+1}$-free graphs on the same vertex set, each of which has the property that every small induced subgraph contains a complete graph on $k$ vertices. The main novelty of our…

Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a…

We prove lower bounds of order $n\log n$ for both the problem to multiply polynomials of degree $n$, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower…

We obtain best possible upper and lower bounds for the Lodge-Ramanujan and DeTemple-Wang approximations to the nth Harmonic Number

We use an old elementary arithmetic argument to find new upper and lower bounds for Sylvester's denumerant function. These bounds are tight enough to get the asymptotic behavior of the denumerant.