Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ramsey numbers of sparse digraphs   Burr and Erd\H{o}s in 1975 conjectured, and Chv\'atal, R\"odl, Szemer\'edi
and Trotter later proved, that the Ramsey number of any bounded degree graph is
linear in the number of vertices. In this paper, we disprove the natural
directed analogue of the Burr--Erd\H{o}s conjecture, answering a question of
Buci\'c, Letzter, and Sudakov. If $H$ is an acyclic digraph, the oriented
Ramsey number of $H$, denoted $\overrightarrow{r_{1}}(H)$, is the least $N$
such that every tournament on $N$ vertices contains a copy of $H$. We show that
for any $\Delta \geq 2$ and any sufficiently large $n$, there exists an acyclic
digraph $H$ with $n$ vertices and maximum degree $\Delta$ such that \[
\overrightarrow{r_{1}}(H)\ge n^{\Omega(\Delta^{2/3}/ \log^{5/3} \Delta)}. \]
This proves that $\overrightarrow{r_{1}}(H)$ is not always linear in the number
of vertices for bounded-degree $H$. On the other hand, we show that
$\overrightarrow{r_{1}}(H)$ is nearly linear in the number of vertices for
typical bounded-degree acyclic digraphs $H$, and obtain linear or nearly linear
bounds for several natural families of bounded-degree acyclic digraphs.
  For multiple colors, we prove a quasi-polynomial upper bound
$\overrightarrow{r_{k}}(H)=2^{(\log n)^{O_{k}(1)}}$ for all bounded-degree
acyclic digraphs $H$ on $n$ vertices, where $\overrightarrow{r_k}(H)$ is the
least $N$ such that every $k$-edge-colored tournament on $N$ vertices contains
a monochromatic copy of $H$. For $k\geq 2$ and $n\geq 4$, we exhibit an acyclic
digraph $H$ with $n$ vertices and maximum degree $3$ such that
$\overrightarrow{r_{k}}(H)\ge n^{\Omega(\log n/\log\log n)}$, showing that
these Ramsey numbers can grow faster than any polynomial in the number of
vertices.

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The categories of this paper are: acc-phys
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The categories of this paper are: chem-ph
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The categories of this paper are: q-bio
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: alg-geom
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The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: econ
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The categories of this paper are: hep-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-ex