Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Defective Ramsey Numbers in Graph Classes   Given a graph $G$, a $k$-sparse $j$-set is a set of $j$ vertices inducing a
subgraph with maximum degree at most $k$. A $k$-dense $i$-set is a set of $i$
vertices that is $k$-sparse in the complement of $G$. As a generalization of
Ramsey numbers, the $k$-defective Ramsey number $R_k^{\mathcal{G}}(i,j)$ for
the graph class $\mathcal{G}$ is defined as the smallest natural number $n$
such that all graphs on $n$ vertices in the class $\mathcal{G}$ have either a
$k$-dense $i$-set or a $k$-sparse $j$-set. In this paper, we examine
$R_k^{\mathcal{G}}(i,j)$ where $\mathcal{G}$ represents various graph classes.
In forests and cographs, we give formulas for all defective Ramsey numbers. In
cacti, bipartite graphs and split graphs, we provide defective Ramsey numbers
in most of the cases and point out open questions, formulated as conjectures if
possible.

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The categories of this paper are: chao-dyn
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The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: alg-geom
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The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
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The categories of this paper are: stat
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The categories of this paper are: physics
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The categories of this paper are: cs
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The categories of this paper are: comp-gas
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The categories of this paper are: bayes-an