Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tight Ramsey bounds for multiple copies of a graph   The Ramsey number $r(G)$ of a graph $G$ is the smallest integer $n$ such that
any $2$ colouring of the edges of a clique on $n$ vertices contains a
monochromatic copy of $G$. Determining the Ramsey number of $G$ is a central
problem of Ramsey theory with long and illustrious history. Despite this there
are precious few classes of graphs $G$ for which the value of $r(G)$ is known
exactly. One such family consists of large vertex disjoint unions of a fixed
graph $H$, we denote such a graph, consisting of $n$ copies of $H$ by $nH$.
This classical result was proved by Burr, Erd\H{o}s and Spencer in 1975, who
showed $r(nH)=(2|H|-\alpha(H))n+c$, for some $c=c(H)$, provided $n$ is large
enough. Since it did not follow from their arguments, Burr, Erd\H{o}s and
Spencer further asked to determine the number of copies we need to take in
order to see this long term behaviour and the value of $c$. More than $30$
years ago Burr gave a way of determining $c(H)$, which only applies when the
number of copies $n$ is triple exponential in $|H|$. In this paper we give an
essentially tight answer to this very old problem of Burr, Erd\H{o}s and
Spencer by showing that the long term behaviour occurs already when the number
of copies is single exponential.

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The categories of this paper are: quant-ph
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The categories of this paper are: q-fin
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: cs
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: eess
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The categories of this paper are: cond-mat