Zeroone klaws and extended zeroone klaws for random distance graphs
Studying zeroone laws for random graphs was initiated by Glebskii Y. et al. in [1]. In this work the authors proved the zeroone law for ErdősRényi random graph G(n,p). Later S. Shelah and J. Spencer expanded the class of functions p(n), for which G(n,p) follows the zeroone law (see [2]). Zeroone laws for random distance graphs have been considered for the rst time by M. Zhukovskii (see [3]). In [4] we studied the zeroone law for a more general model of random distance graphs.
Let {G_{n} = (V_{n}, E_{n})}^{∞}_{n=1} be a a sequence of distance graphs and p = p(n) be a function of n. The random distance graph G(G_{n},p) is the probabilistic space (Ω_{Gn},F_{Gn},Ρ_{Gnρ}), where
Ω_{Gn} = {G = (V,E) : V = V_{n}, E ⊆ E_{n}},
F_{Gn} = 2 ^{ΩGn}, F_{Gn,p}(G) = p^{E}(1p)^{EnE}.
We say sequence G(G_{n}, p) follows zeroone klaw if for any firstorder property L with quantier depth at most k the probability P_{Gn ,p}(L) of the event "G(G_{n} ,p) possesses property L" tends either to 0 or to 1 as n → ∞. We say sequence G(G_{n}, p) follows extended zeroone klaw if for any firstorder property L with quantier depth at most k any partial limit of the sequence {P_{Gn ,p}(L)}^{∞}_{n=1} equals either 0 or 1.
We obtain conditions on the sequence {G_{n}}^{∞}_{n=1} under which one of the following three mutually exclusive cases occurs:

zeroone klaw holds;

zeroone klaw doesn't hold, but extended zeroone klaw holds;

extended zeroone klaw doesn't hold.
References
[1] Y.V. Glebskii, D.I. Kogan, M.I. Liagonkii, V.A. Talanov, Range and degree of realizability of formulas the restricted predicate calculus, Cybernetics 5: 142154,1969. (Russian original: Kibernetica 5, 1727)
[2] S. Shelah, J.H. Spencer, Zeroone laws for sparse random graphs, J. Amer. Math. Soc. 1: 97115, 1988.
[3] M.E. Zhukovskii, On a sequence of random distance graphs subject to the zeroone law, Problems of Information Transmission 47(3): 251268, 2011. (Russian original: Problemy Peredachi Informatsii, 47(3): 3958, 2011).
[4] S.N. Popova, Zeroone law for random distance graphs with vertices in {1; 0; 1}^{n}, Problems of Information Transmission 50(1), 2014. (Russian original: Problemy Peredachi Informatsii, 50(1): 79101, 2014).