Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Constrained Ramsey Numbers   For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum
n such that every edge coloring of the complete graph on n vertices, with any
number of colors, has a monochromatic subgraph isomorphic to S or a rainbow
(all edges differently colored) subgraph isomorphic to T. The Erdos-Rado
Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star
or T is acyclic, and much work has been done to determine the rate of growth of
f(S, T) for various types of parameters. When S and T are both trees having s
and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <=
O(st^2) and conjectured that it is always at most O(st). They also mentioned
that one of the most interesting open special cases is when T is a path. In
this work, we study this case and show that f(S, P_t) = O(st log t), which
differs only by a logarithmic factor from the conjecture. This substantially
improves the previous bounds for most values of s and t.

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The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph