Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Refined Tur\'an numbers and Ramsey numbers for the loose 3-uniform path
  of length three   Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices
$a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is
known that the $r$-color Ramsey number for $P$ is $R(P;r)=r+6$ for $r\le 7$.
The proof of this result relies on a careful analysis of the Tur\'an numbers
for $P$. In this paper, we refine this analysis further and compute, for all
$n$, the third and fourth order Tur\'an numbers for $P$. With the help of the
former, we confirm the formula $R(P;r)=r+6$ for $r\in\{8,9\}$.

28
The categories of this paper are: alg-geom
32
The categories of this paper are: atom-ph
19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
9
The categories of this paper are: nlin
17
The categories of this paper are: q-fin
11
The categories of this paper are: hep-lat
1
The categories of this paper are: cs
16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph