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]]>is within of

for any pair of 01-valued functions .

If we now take the expression

and apply that assumption to the inner expectation for each fixed , we find that we can approximate the whole expression to within by

The is because we get equality when and equality to within otherwise.

And now we can replace the by at the cost of another contribution of to the error and we get what we want.

So what I should have said is that this particular formulation of the regularity property lends itself quite conveniently to proving a counting lemma — you just repeatedly replace edges by s, so to speak, and you don’t have to say anything about most vertices having roughly the expected degree.

]]>But clearly if you are trying to prove Property 2 you only need to prove it for X and Y.

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