Indistinguishable families of graphs

Is there some natural not too trivial graph property such that all its
members are look-alikes. More precisely:

Can we find set of graphs S such that for any k there is N satisfying:
any two graphs of order larger than N satisfy the same first-order
formulas with at most k quantifiers, or equivalently are
undistinguashable by a k-round Ehrefeucht-Fraisse game?

.

Relevant Pages

  • Re: Indistinguishable families of graphs
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  • Re: Question on computational complexity
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  • Re: Indistinguishable families of graphs
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