Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rolling backwards can move you forward: on embedding problems in sparse
  expanders   We develop a general embedding method based on the Friedman-Pippenger tree
embedding technique (1987) and its algorithmic version, essentially due to
Aggarwal et al. (1996), enhanced with a roll-back idea allowing to sequentially
retrace previously performed embedding steps. We use this method to obtain the
following results.
  -We show that the size-Ramsey number of logarithmically long subdivisions of
bounded degree graphs is linear in their number of vertices, settling a
conjecture of Pak (2002).
  -We give a deterministic, polynomial time online algorithm for finding
vertex-disjoint paths of prescribed length between given pairs of vertices in
an expander graph. Our result answers a question of Alon and Capalbo (2007).
  -We show that relatively weak bounds on the spectral ratio of $d$-regular
graphs force the existence of a topological minor of $K_t$ where $t=(1-o(1))d$.
We also exhibit a construction which shows that the theoretical maximum $t=d+1$
cannot be attained even if $\lambda=O(\sqrt{d})$. This answers a question of
Fountoulakis, K\"uhn and Osthus (2009).

9
The categories of this paper are: nlin
16
The categories of this paper are: stat
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
7
The categories of this paper are: hep-th
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
11
The categories of this paper are: hep-lat
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
21
The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
23
The categories of this paper are: solv-int