Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rainbow clique subdivisions and blow-ups   We show that for every integer $m \ge 2$ and large $n$, every properly
edge-coloured graph on $n$ vertices with at least $n (\log n)^{60}$ edges
contains a rainbow subdivision of $K_m$. Using the same framework, we also
prove a result about the extremal number of $r$-blow-ups of subdivisions. We
show that for integers $r, m \ge 2$ and large $n$, every graph on $n$ vertices
with at least $n^{2 - \frac{1}{r}} (\log n)^{\frac{60}{r}}$ edges has an
$r$-blow-up of a subdivision of $K_m$. Both results are sharp up to a
polylogarithmic factor. Our proofs use the connection between mixing time of
random walks and expansion in graphs.

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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: econ
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The categories of this paper are: hep-lat
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The categories of this paper are: mtrl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: supr-con
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The categories of this paper are: eess
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The categories of this paper are: nlin
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The categories of this paper are: cond-mat
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The categories of this paper are: plasm-ph