Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Kruskal-Katona type Problem   The Kruskal Katona theorem was proved in the 1960s. In the theorem, we are
given an integer $r$ and families of sets $\mathcal{A}\subset \mathbb{N}^{(r)}$
and $\mathcal{B}\subset\mathbb{N}^{(r-1)}$ such that for every
$A\in\mathcal{A}$, every subset of $A$ of size $r-1$ is in $\mathcal{B}$. We
are interested in finding the mimimum size of $b=|\mathcal{B}|$ given fixed
values of $r$ and $a=|\mathcal{A}|$. The Kruskal Katona theorem states that
this mimimum occurs when both $\mathcal{A}$ and $\mathcal{B}$ are initial
segments of the colexicographic ordering. The Kruskal Katona theorem is very
useful and has had many applications and generalisations.
  In this paper, we are interested in one particular generalisation, where
instead of every subset of $A$ of size $r-1$ being in $\mathcal{B}$, we will
instead ask that only $k$ of them are, where $k$ is some integer smaller than
$r$. Note that setting $k=r$ is exactly the Kruskal Katona theorem. We will
first find exact results for the cases where $0\leq k \leq 3$. For $k\geq 4$ we
will not solve the question completely, however, we will find the exact result
for infinitely many $a$. We will also provide a formula that is within some
additive constant of the correct result for all $a$.

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The categories of this paper are: q-bio
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The categories of this paper are: math-ph
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The categories of this paper are: stat
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The categories of this paper are: cond-mat
1
The categories of this paper are: cs
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: plasm-ph
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
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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: supr-con
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-th