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Oct 24, 2014 at 11:56 comment added Ilya Bogdanov One may give a better lower bound for $N(n,m,k;1)$ if $n$, $m$, and $k$ are large enough. Namely, one may fill all the `planes' with even first coordinate and choose an arbitrary nonempty subset in any plane with odd first coordinate. This way you obtain $(2^{mk}-1)^{\lceil n/2\rceil}$ connected sets.
Oct 5, 2014 at 16:20 comment added user40780 Sure:) I will contact you when I have other ideas:) Thank you
Oct 5, 2014 at 16:16 comment added Joonas Ilmavirta You are welcome! If you want to discuss possible collaboration, send email. There is a link in my profile to my homepage.
Oct 5, 2014 at 16:12 comment added user40780 If you are interested, we could have further collaborations to write some papers:)
Oct 5, 2014 at 16:05 comment added user40780 great! Thank you:) For the material science problem I am facing, I will try to define a tighter way than connectedness to create the bound...
Oct 5, 2014 at 16:01 vote accept user40780
Oct 5, 2014 at 14:32 history edited Joonas Ilmavirta CC BY-SA 3.0
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Oct 5, 2014 at 9:35 history answered Joonas Ilmavirta CC BY-SA 3.0