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10 votes
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$3\times 3$ magic squares consisting of entries of a dense set $D\subseteq \mathbb{N}$

Yes. By Szemerédi's theorem, your set contains an arithmetic progression of arbitrary length. In particular, it contains a progression of length 9, say it's $d_1,\ldots,d_9$. Then $$ \begin{pmatrix} ...
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17 votes
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Page-turning number of a graph

The page-turning number of a graph $G$ is also known as the bandwidth of $G$ (https://en.wikipedia.org/wiki/Graph_bandwidth). The Wikipedia page also contains values of the bandwidth for some special ...
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1 vote
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Two dice yielding uniform distribution, part 2

Suppose that you can factor $x^{p-1} + x^{p-2} + \cdots + 1$ as $f(x)g(x)$ with $\deg(f)=a,\deg(g) = b$ and $f,g$ having nonnegative coefficients. Lemma: It must be the case that $f = \sum_{i=0}^a c_i ...
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