Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On minimal Ramsey graphs and Ramsey equivalence in multiple colours   For an integer $q\ge 2$, a graph $G$ is called $q$-Ramsey for a graph $H$ if
every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$.
If $G$ is $q$-Ramsey for $H$, yet no proper subgraph of $G$ has this property
then $G$ is called $q$-Ramsey-minimal for $H$. Generalising a statement by
Burr, Ne\v{s}et\v{r}il and R\"odl from 1977 we prove that, for $q\ge 3$, if $G$
is a graph that is not $q$-Ramsey for some graph $H$ then $G$ is contained as
an induced subgraph in an infinite number of $q$-Ramsey-minimal graphs for $H$,
as long as $H$ is $3$-connected or isomorphic to the triangle. For such $H$,
the following are some consequences.
  (1) For $2\le r< q$, every $r$-Ramsey-minimal graph for $H$ is contained as
an induced subgraph in an infinite number of $q$-Ramsey-minimal graphs for $H$.
  (2) For every $q\ge 3$, there are $q$-Ramsey-minimal graphs for $H$ of
arbitrarily large maximum degree, genus, and chromatic number.
  (3) The collection $\{{\cal M}_q(H) : H \text{ is 3-connected or } K_3\}$
forms an antichain with respect to the subset relation, where ${\cal M}_q(H)$
denotes the set of all graphs that are $q$-Ramsey-minimal for $H$.
  We also address the question which pairs of graphs satisfy ${\cal
M}_q(H_1)={\cal M}_q(H_2)$, in which case $H_1$ and $H_2$ are called
$q$-equivalent. We show that two graphs $H_1$ and $H_2$ are $q$-equivalent for
even $q$ if they are $2$-equivalent, and that in general $q$-equivalence for
some $q\ge 3$ does not necessarily imply $2$-equivalence. Finally we indicate
that for connected graphs this implication may hold: Results by
Ne\v{s}et\v{r}il and R\"odl and by Fox, Grinshpun, Liebenau, Person and Szab\'o
imply that the complete graph is not $2$-equivalent to any other connected
graph. We prove that this is the case for an arbitrary number of colours.

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The categories of this paper are: nucl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: physics
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The categories of this paper are: math-ph
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: stat
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The categories of this paper are: acc-phys
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: hep-ph