Geodesic
On the 4-spectrum of first-order properties of random graphs
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 500 (2021), pp. 31-34

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A $k$-spectrum is a set of all positive $\alpha$ such that the random binomial graph $G(n,n^{-\alpha})$ does not obey the zero–one law for first-order formulas with a quantifier depth at most $k$. We have proved that the minimal $k$ such that the $k$-spectrum is infinite equals 5.
Keywords: first-order logic, random binomial graph, zero–one law, spectrum of formula, Ehrenfeucht–Fraïssé game. 
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     author = {M. E. Zhukovskii and A. D. Matushkin and Yu. N. Yarovikov},
     title = {On the 4-spectrum of first-order properties of random graphs},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
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     publisher = {mathdoc},
     volume = {500},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2021_500_a5/}
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M. E. Zhukovskii; A. D. Matushkin; Yu. N. Yarovikov. On the 4-spectrum of first-order properties of random graphs. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 500 (2021), pp. 31-34. http://geodesic.mathdoc.fr/item/DANMA_2021_500_a5/