Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniquely $K^{(k)}_r$-saturated Hypergraphs   In this paper we generalize the concept of uniquely $K_r$-saturated graphs to
hypergraphs. Let $K_r^{(k)}$ denote the complete $k$-uniform hypergraph on $r$
vertices. For integers $k,r,n$ such that $2\le k <r<n$, a $k$-uniform
hypergraph $H$ with $n$ vertices is uniquely $K_r^{(k)}$-saturated if $H$ does
not contain $K_r^{(k)}$ but adding to $H$ any $k$-set that is not a hyperedge
of $H$ results in exactly one copy of $K_r^{(k)}$. Among uniquely
$K_r^{(k)}$-saturated hypergraphs, the interesting ones are the primitive ones
that do not have a dominating vertex---a vertex belonging to all possible
${n-1\choose k-1}$ edges. Translating the concept to the complements of these
hypergraphs, we obtain a natural restriction of $\tau$-critical hypergraphs: a
hypergraph $H$ is uniquely $\tau$-critical if for every edge $e$,
$\tau(H-e)=\tau(H)-1$ and $H-e$ has a unique transversal of size $\tau(H)-1$.
  We have two constructions for primitive uniquely $K_r^{(k)}$-saturated
hypergraphs. One shows that for $k$ and $r$ where $4\le k<r\le 2k-3$, there
exists such a hypergraph for every $n>r$. This is in contrast to the case $k=2$
and $r=3$ where only the Moore graphs of diameter two have this property. Our
other construction keeps $n-r$ fixed; in this case we show that for any fixed
$k\ge 2$ there can only be finitely many examples. We give a range for $n$
where these hypergraphs exist. For $n-r=1$ the range is completely determined:
$k+1\le n \le {(k+2)^2\over 4}$. For larger values of $n-r$ the upper end of
our range reaches approximately half of its upper bound. The lower end depends
on the chromatic number of certain Johnson graphs.

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The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ex
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The categories of this paper are: cs
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-th