Maksim Zhukovskii

First order sentences about random graphs: Small number of alternations

Spectrum of a first order sentence is the set of all

The Descriptive Complexity of Subgraph Isomorphism Without Numerics

alpha

On the First-Order Complexity of Induced Subgraph Isomorphism.

such that

Spectra of formulas with bounded quantifier alternations

G(n, n^{-alpha})

Tight Bounds on the Asymptotic Descriptive Complexity of Subgraph Isomorphism.

does not obey zero-one law w.r.t. this sentence. We have proved that the minimal number of quantifier alternations of a first order sentence with an infinite spectrum equals 3. We have also proved that the spectrum of a first order sentence with a quantifier depth 4 has no limit points except possibly the points 1/2 and 3/5.

On a generalization of a Ramanujan conjecture for binomial random variables

Let F be a connected graph with (ell ) vertices. The existence of a subgraph isomorphic to F can be defined in first-order logic with quantifier depth no better than (ell ), simply because no first-order formula of smaller quantifier depth can distinguish between the complete graphs (K_ell ) and (K_{ell -1}). We show that, for some F, the existence of an F subgraph in sufficiently large connected graphs is definable with quantifier depth (ell -3). On the other hand, this is never possible with quantifier depth better than (ell /2). If we, however, consider definitions over connected graphs with sufficiently large treewidth, the quantifier depth can for some F be arbitrarily small comparing to (ell ) but never smaller than the treewidth of F.