Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: How Expressive Are Friendly School Partitions?   A natural procedure for assigning students to classes in the beginning of the
school-year is to let each student write down a list of $d$ other students with
whom she/he wants to be in the same class (typically $d=3$). The teachers then
gather all the lists and try to assign the students to classes in a way that
each student is assigned to the same class with at least one student from
her/his list. We refer to such partitions as friendly. In realistic scenarios,
the teachers may also consider other constraints when picking the friendly
partition: e.g. there may be a group of students whom the teachers wish to
avoid assigning to the same class; alternatively, there may be two close
friends whom the teachers want to put together; etc.
  Inspired by such challenges, we explore questions concerning the
expressiveness of friendly partitions. For example: Does there always exist a
friendly partition? More generally, how many friendly partitions are there? Can
every student $u$ be separated from any other student $v$? Does there exist a
student $u$ that can be separated from any other student $v$?
  We show that when $d\geq 3$ there always exist at least $2$ friendly
partitions and when $d\geq 15$ there always exists a student $u$ which can be
separated from any other student $v$. The question regarding separability of
each pair of students is left open, but we give a positive answer under the
additional assumption that each student appears in at most roughly $\exp(d)$
lists. We further suggest several open questions and present some preliminary
findings towards resolving them.

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The categories of this paper are: acc-phys
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The categories of this paper are: cond-mat
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nlin
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The categories of this paper are: bayes-an
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-th