Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ramsey numbers of 3-uniform loose paths and loose cycles   Haxell et. al. [%P. Haxell, T. Luczak, Y. Peng, V. R\"{o}dl, A.
%Ruci\'{n}ski, M. Simonovits, J. Skokan, The Ramsey number for hypergraph
cycles I, J. Combin. Theory, Ser. A, 113 (2006), 67-83] proved that the 2-color
Ramsey number of 3-uniform loose cycles on $2n$ vertices is asymptotically
$\frac{5n}{2}$. Their proof is based on the method of Regularity Lemma. Here,
without using this method, we generalize their result by determining the exact
values of 2-color Ramsey numbers involving loose paths and cycles in 3-uniform
hypergraphs. More precisely, we prove that for every $n\geq m\geq 3$,
$R(\mathcal{P}^3_n,\mathcal{P}^3_m)=R(\mathcal{P}^3_n,\mathcal{C}^3_m)=R(\mathcal{C}^3_n,\mathcal{C}^3_m)+1=2n+\lfloor\frac{m+1}{2}\rfloor$
and for $n>m\geq3$,
$R(\mathcal{P}^3_m,\mathcal{C}^3_n)=2n+\lfloor\frac{m-1}{2}\rfloor$. These give
a positive answer to a question of Gy\'{a}rf\'{a}s and Raeisi [The Ramsey
number of loose triangles and quadrangles in hypergraphs, Electron. J. Combin.
19 (2012), #R30].

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The categories of this paper are: dg-ga
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin
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The categories of this paper are: hep-th
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: funct-an
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The categories of this paper are: cs
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The categories of this paper are: eess
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The categories of this paper are: supr-con
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The categories of this paper are: chem-ph