Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Exchange properties of finite set-systems   In a recent breakthrough, Adiprasito, Avvakumov, and Karasev constructed a
triangulation of the $n$-dimensional real projective space with a
subexponential number of vertices. They reduced the problem to finding a small
downward closed set-system $\cal F$ covering an $n$-element ground set which
satisfies the following condition: For any two disjoint members $A, B\in\cal
F$, there exist $a\in A$ and $b\in B$ such that either $B\cup\{a\}\in\cal F$
and $A\cup\{b\}\setminus\{a\}\in\cal F$, or $A\cup\{b\}\in\cal F$ and
$B\cup\{a\}\setminus\{b\}\in\cal F$. Denoting by $f(n)$ the smallest
cardinality of such a family $\cal F$, they proved that $f(n)<2^{O(\sqrt{n}\log
n)}$, and they asked for a nontrivial lower bound. It turns out that the
construction of Adiprasito et al. is not far from optimal; we show that
$2^{(1.42+o(1))\sqrt{n}}\le f(n)\le 2^{(1+o(1))\sqrt{2n\log n}}$.
  We also study a variant of the above problem, where the condition is
strengthened by also requiring that for any two disjoint members $A, B\in\cal
F$ with $|A|>|B|$, there exists $a\in A$ such that $B\cup\{a\}\in\cal F$. In
this case, we prove that the size of the smallest $\cal F$ satisfying this
stronger condition lies between $2^{\Omega(\sqrt{n}\log n)}$ and
$2^{O(n\log\log n/\log n)}$.

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The categories of this paper are: math-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: hep-lat
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: q-fin
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The categories of this paper are: stat
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The categories of this paper are: atom-ph