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{\bf A. Czygrinow and B. Nagle}
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{\bf Matrix-Free Proof of a Regularity Characterization}
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The central concept in Szemer\'edi's powerful regularity lemma is the
so-called $\epsilon$-regular pair.  A useful statement of Alon {\it et al.\/}
essentially equates the notion of an $\epsilon$-regular pair with degree
uniformity of vertices and pairs of vertices.  The known proof of this
characterization uses a clever matrix argument.


This paper gives a simple proof of the characterization without
appealing to the matrix argument of Alon {\it et al}. We show the
$\epsilon$-regular characterization follows from an application of
Szemer\'edi's regularity lemma itself.

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