An optimal lower bound on the number of variables for graph identification

It is shown that Omega (n) variables are needed for first-order logic with counting to identify graphs on n vertices. This settles a long-standing open problem. The lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that three variables suffice to identify all graphs of color class size 3, and two variables suffice to identify almost all graphs. The lower bound is optimal up to multiplication by a constant because n variables obviously suffice to identify graphs on n vertices.<
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