Related papers: Ramsey numbers and adiabatic quantum computing
This paper sets out the results of a range of searches for linear and cyclic graph colourings with specific Ramsey properties. The new graphs comprise mainly 'template graphs' which can be used in a construction described by the current…
A construction described by the current author (2017) uses two linear prototypes to build a compound graph with Ramsey properties inherited from the prototype graphs. The resulting graph is linear; and cyclic if both prototypes are cyclic.…
We study an architecture for implementing adiabatic quantum computation with trapped neutral atoms. Ground state atoms are dressed by laser fields in a manner conditional on the Rydberg blockade mechanism, thereby providing the requisite…
We both survey and extend a new technique from Lu Liu to prove separation theorems between products of Ramsey-type theorems over computable reducibility. We use this technique to show that Ramsey's theorem for $n$-tuples and three colors is…
We show enough evidence that a structured version of Adiabatic Quantum Computation (AQC) is efficient for most satisfiability problems. More precisely, when the success probability is fixed beforehand, the computational resources grow…
The Ramsey numbers $R(T_n,W_8)$ are determined for each tree graph $T_n$ of order $n\geq 7$ and maximum degree $\Delta(T_n)$ equal to either $n-4$ or $n-5$. These numbers indicate strong support for the conjecture, due to Chen, Zhang and…
Let $K_m^{(3)}$ denote the complete $3$-uniform hypergraph on $m$ vertices and $S_n^{(3)}$ the $3$-uniform hypergraph on $n+1$ vertices consisting of all $\binom{n}{2}$ edges incident to a given vertex. Whereas many hypergraph Ramsey…
The $q$-color Ramsey number of a $k$-uniform hypergraph $H$ is the minimum integer $N$ such that any $q$-coloring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. The study of these numbers is one…
We establish new lower bounds for $28$ classical two and three color Ramsey numbers, and describe the heuristic search procedures we used. Several of the new three color bounds are derived from the two color constructions; specifically, we…
Let $n\geq\nu$, let $T$ be an $n$-vertex tree with bipartition class sizes $t_1\geq t_2$, and let $S$ be a $\nu$-vertex tree with bipartition class sizes $\tau_1\geq\tau_2$. Using four natural constructions, we show that the Ramsey number…
We review methods for coherently controlling Rydberg quantum states of atomic ensembles using Adiabatic Rapid Passage and Stimulated Raman Adiabatic Passage. These methods are commonly used for population inversion in simple two-level and…
The upper (mixed) domination Ramsey number $u(m, n)$($v(m,n)$) is the smallest integer $p$ such that every $2$-coloring of the edges of $K_p$ with color red and blue, $\Gamma(B) \geq m$ or $\Gamma(R) \geq n$ ($\beta(R) \geq n$); where $B$…
Quantum computing for machine learning attracts increasing attention and recent technological developments suggest that especially adiabatic quantum computing may soon be of practical interest. In this paper, we therefore consider this…
Given graphs $G, H$ and an integer $q \ge 2$, the generalized Ramsey number, denoted $r(G,H,q)$, is the minimum number of colours needed to edge-colour $G$ such that every copy of $H$ receives at least $q$ colours. In this paper, we prove…
In a seminal paper from 1983, Burr and Erdos started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the…
Besides the traditional circuit-based model of quantum computation, several quantum algorithms based on a continuous-time Hamiltonian evolution have recently been introduced, including for instance continuous-time quantum walk algorithms as…
Assume that $K_{j\times n}$ be a complete, multipartite graph consisting of $j$ partite sets and $n$ vertices in each partite set. For given graphs $G_1, G_2,\ldots, G_n$, the multipartite Ramsey number (M-R-number) $m_j(G_1, G_2,…
We present an efficient quantum algorithm for some independent set problems in graph theory, based on non-abelian adiabatic mixing. We illustrate the performance of our algorithm with analysis and numerical calculations for two different…
In this note, we investigate for various pairs of graphs $(H,G)$ the question of how many random edges must be added to a dense graph to guarantee that any red-blue coloring of the edges contains a red copy of $H$ or a blue copy of $G$. We…
The set-colouring Ramsey number $R_{r,s}(k)$ is defined to be the minimum $n$ such that if each edge of the complete graph $K_n$ is assigned a set of $s$ colours from $\{1,\ldots,r\}$, then one of the colours contains a monochromatic clique…