Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the size of spheres of relations with a transitive group of
  automorphisms   Let $\Gamma =(V,E)$ be a point-transitive reflexive relation. Let $v\in V$
and put $r=|\Gamma (v)|.$ Also assume $\Gamma ^j(v)\cap \Gamma ^{-}(v)=\{v\}$.
Then $$ |\Gamma ^{j} (v)\setminus \Gamma ^{j-1} (v)| \ge r-1.$$
  In particular we have $ |\Gamma ^{j} (v)| \ge 1+(r-1)j.$ The last result
confirms a recent conjecture of Seymour in the case vertex-transitive graphs.
Also it gives a short proof for the validity of the Caccetta-H\"aggkvist
conjecture for vertex-transitive graphs and generalizes an additive result of
Shepherdson.

12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
28
The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
13
The categories of this paper are: quant-ph
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con