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Probabilistic statement on matrix ranks
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Probabilistic statement on matrix ranks
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accepted  Probabilistic statement on matrix ranks 
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Probabilistic statement on matrix ranks
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Constructing a particular Cayley Graph
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comment 
Constructing a particular Cayley Graph
Sorry there were mistakes as pointed by Johannes Hahn. I corrected them. Question is stable now. Thank you for your comments. Made a general observation from where question came from/motivation. 
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Constructing a particular Cayley Graph
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Probabilistic statement on matrix ranks
You are saying that: $$\forall r>0,c>0\exists n\in \Bbb N, A\in\{0,1\}^{n\times n}\mbox{ with }rank(A)=r: \frac{n}{2\log_2n}\ll\frac{r^{({\log_2r})^c}}{2\log_2r}?$$ This gives: $$1_n'A1_n<\frac{rn}{2\log_2n}\ll\frac{(r1)r^{({\log_2r})^c}}{2\log_2r}?$$ 
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Constructing a particular Cayley Graph
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comment 
Constructing a particular Cayley Graph
$M$ depends on $P$ to parametrize everything by $P$. 
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Constructing a particular Cayley Graph
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