Maksim Zhukovskii – Zero-one k-laws for G(n,n−α)

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We study asymptotical behavior of the probabilities of first-order properties for Erdős-Rényi random graphs G(n,p(n)) with p(n)=n-α, α ∈ (0,1). The following zero-one law was proved in 1988 by S. …

We study asymptotical behavior of the probabilities of first-order properties for Erdős-Rényi random graphs G(n,p(n)) with p(n)=n-α, α ∈ (0,1). The following zero-one law was proved in 1988 by S. Shelah and J.H. Spencer [1]: if α is irrational then for any first-order property L either the random graph satisfies the property L asymptotically almost surely or it doesn't satisfy (in such cases the random graph is said to obey zero-one law. When α ∈ (0,1) is rational the zero-one law for these graphs doesn't hold.

Let k be a positive integer. Denote by Lk the class of the first-order properties of graphs defined by formulae with quantifier depth bounded by the number k (the sentences are of a finite length). Let us say that the random graph obeys zero-one k-law, if for any first-order property L ∈ Lk either the random graph satisfies the property L almost surely or it doesn't satisfy. Since 2010 we prove several zero-one $k$-laws for rational α from Ik=(0, 1/(k-2)] ∪ [1-1/(2k-1), 1). For some points from Ik we disprove the law. In particular, for α ∈ (0, 1/(k-2)) ∪ (1-1/2k-2, 1) zero-one k-law holds. If α ∈ {1/(k-2), 1-1/(2k-2)}, then zero-one law does not hold (in such cases we call the number α k-critical).

We also disprove the law for some α ∈ [2/(k-1), k/(k+1)]. From our results it follows that zero-one 3-law holds for any α ∈ (0,1). Therefore, there are no 3-critical points in (0,1). Zero-one 4-law holds when α ∈ (0,1/2) ∪ (13/14,1). Numbers 1/2 and 13/14 are 4-critical. Moreover, we know some rational 4-critical and not 4-critical numbers in [7/8,13/14). The number 2/3 is 4-critical. Recently we obtain new results concerning zero-one 4-laws for the neighborhood of the number 2/3.

References

[1] S. Shelah, J.H. Spencer, Zero-one laws for sparse random graphs, J. Amer. Math. Soc.

1: 97–115, 1988.

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  • 1. Zero-one k-laws and extended zero-one k-laws for random distance graphs Popova Svetlana vomonosov wos™ow ƒt—te …niversity ‡orkshop on ixtrem—l qr—ph „heory wos™owD tune TD PHIR 1/22
  • 2. he(nitionsX ird¥osE‚¡enyi r—ndom gr—ph G(n, p) —nd r—ndom gr—ph G(Gn, p) he(nitionF ird¥osE‚¡enyi r—ndom gr—ph G(n, p) is — r—ndom element with v—lues in Ωn —nd distri˜ution Pn,p on FnD where Ωn = {(V = {1, ..., n}, E)}, Fn = 2Ωn , Pn,p(G) = p|E| (1 − p)C2 n−|E| . 2/22
  • 3. he(nitionsX ird¥osE‚¡enyi r—ndom gr—ph G(n, p) —nd r—ndom gr—ph G(Gn, p) he(nitionF ird¥osE‚¡enyi r—ndom gr—ph G(n, p) is — r—ndom element with v—lues in Ωn —nd distri˜ution Pn,p on FnD where Ωn = {(V = {1, ..., n}, E)}, Fn = 2Ωn , Pn,p(G) = p|E| (1 − p)C2 n−|E| . he(nitionF vet Gn ˜e — sequen™e of gr—phs Gn = (Vn, En)F ‚—ndom gr—ph G(Gn, p) is — r—ndom element with v—lues in ΩGn —nd distri˜ution PGn,p on FGn D where ΩGn = {G = (V, E) : V = Vn, E ⊆ En}, FGn = 2ΩGn , PGn,p(G) = p|E| (1 − p)|En|−|E| . 2/22
  • 4. he(nitionsX (rstEorder properties —nd zeroEone l—w he(nitionF pirstEorder properties of gr—phs —re de(ned ˜y (rstEorder formul—eD whi™h —re ˜uilt of predi™—te sym˜ols ∼, = logi™—l ™onne™tivities ¬, ⇒, ⇔, ∨, ∧ v—ri—˜les x, y, . . . qu—nti(ers ∀, ∃ 3/22
  • 5. he(nitionsX (rstEorder properties —nd zeroEone l—w he(nitionF pirstEorder properties of gr—phs —re de(ned ˜y (rstEorder formul—eD whi™h —re ˜uilt of predi™—te sym˜ols ∼, = logi™—l ™onne™tivities ¬, ⇒, ⇔, ∨, ∧ v—ri—˜les x, y, . . . qu—nti(ers ∀, ∃ he(nitionF „he r—ndom gr—ph G(n, p) is s—id to follow zeroEone l—w if for —ny (rstEorder property L either lim n→∞ Pn,p(L) = 0 or lim n→∞ Pn,p(L) = 1. 3/22
  • 6. he(nitionsX zeroEone kEl—w he(nitionF „he r—ndom gr—ph G(n, p) is s—id to follow zeroEone kEl—w if for —ny property L de(ned ˜y — (rstEorder formul— with qu—nti(er depth —t most k either lim n→∞ Pn,p(L) = 0 or lim n→∞ Pn,p(L) = 1. 4/22
  • 7. eroEone l—w for ird¥osE‚¡enyi r—ndom gr—ph G(n, p) „heorem@qle˜ski et —lFD IWTWY p—ginD IWUTA Let a function p = p(n) satisfy the property ∀β > 0 min(p, 1 − p)nβ → ∞ when n → ∞. Then the random graph G(n, p) follows the zero-one law. 5/22
  • 8. eroEone l—w for ird¥osE‚¡enyi r—ndom gr—ph G(n, p) „heorem@qle˜ski et —lFD IWTWY p—ginD IWUTA Let a function p = p(n) satisfy the property ∀β > 0 min(p, 1 − p)nβ → ∞ when n → ∞. Then the random graph G(n, p) follows the zero-one law. „heorem@ƒhel—hD ƒpen™erD IWVVA Let p(n) = n−β and β be an irrational number, 0 < β < 1. Then the random graph G(n, p) follows the zero-one law. 5/22
  • 9. ‚—ndom dist—n™e gr—ph ‚—ndom dist—n™e gr—ph G(Gdist n , p) Gdist n = (V dist n , Edist n ) a = a(n), c = c(n) V dist n = v = (v1 , . . . , vn ) : vi ∈ {0, 1}, n i=1 vi = a Edist n = {{u, v} ∈ V dist n × V dist n : (u, v) = c} 6/22
  • 10. eroEone l—w for r—ndom dist—n™e gr—ph vet — fun™tion p = p(n) s—tisfy the property ∀β > 0 min(p, 1 − p)|V dist n |β → ∞ when n → ∞. 7/22
  • 11. eroEone l—w for r—ndom dist—n™e gr—ph vet — fun™tion p = p(n) s—tisfy the property ∀β > 0 min(p, 1 − p)|V dist n |β → ∞ when n → ∞. „heorem Let a(n) = αn, c(n) = α2n, α ∈ Q, 0 < α < 1. Then the random graph G(Gdist n , p) doesn't follow the zero-one law, but there exists a subsequence G(Gdist ni , p) following the zero-one law. 7/22
  • 12. uestions ‡hen does — given su˜sequen™e G(Gdist ni , p) follow zeroEone l—wc 8/22
  • 13. uestions ‡hen does — given su˜sequen™e G(Gdist ni , p) follow zeroEone l—wc hoes there exist — (rstEorder property L —nd — su˜sequen™e G(Gdist ni , p) su™h th—t lim i→∞ PGdist ni ,p(L) ∈ (0, 1) 8/22
  • 14. uestions ‡hen does — given su˜sequen™e G(Gdist ni , p) follow zeroEone l—wc hoes there exist — (rstEorder property L —nd — su˜sequen™e G(Gdist ni , p) su™h th—t lim i→∞ PGdist ni ,p(L) ∈ (0, 1) ‡h—t limiting pro˜—˜ilities PGdist ni ,p(L) ™—n we getc 8/22
  • 15. ixtended zeroEone kEl—w he(nitionF „he r—ndom gr—ph G(Gn, p) is s—id to follow extended zeroEone kEl—w if for every property L de(ned ˜y — (rstEorder formul— with qu—nti(er depth —t most k —ny p—rti—l limit of the sequen™e PGn,p(L) equ—ls either 0 or 1F 9/22
  • 16. ixtended zeroEone kEl—w he(nitionF „he r—ndom gr—ph G(Gn, p) is s—id to follow extended zeroEone kEl—w if for every property L de(ned ˜y — (rstEorder formul— with qu—nti(er depth —t most k —ny p—rti—l limit of the sequen™e PGn,p(L) equ—ls either 0 or 1F qo—lF pind ™onditions on the sequen™e G(Gdist ni , p) under whi™h one of the following t—kes pl—™eX zeroEone kEl—w holds zeroEone kEl—w doesn9t holdD ˜ut extended zeroEone kEl—w holds extended zeroEone kEl—w doesn9t hold 9/22
  • 17. ihrenfeu™ht g—me EHR(G, H, k) EHR(G, H, k) qr—phs G, HD num˜er of rounds k „wo pl—yers ƒpoiler —nd hupli™—tor iEth roundX ƒpoiler ™hooses — vertex either from G or from H hupli™—tor ™hooses — vertex of the other gr—ph vet x1, . . . , xkD y1, . . . , yk ˜e verti™es ™hosen from gr—phs G —nd H respe™tivelyF hupli™—tor wins if —nd only if G|{x1,...,xk} ∼= H|{y1,...,yk}F 10/22
  • 18. ihrenfeu™ht g—me EHR(G, H, k) EHR(G, H, k) qr—phs G, HD num˜er of rounds k „wo pl—yers ƒpoiler —nd hupli™—tor iEth roundX ƒpoiler ™hooses — vertex either from G or from H hupli™—tor ™hooses — vertex of the other gr—ph vet x1, . . . , xkD y1, . . . , yk ˜e verti™es ™hosen from gr—phs G —nd H respe™tivelyF hupli™—tor wins if —nd only if G|{x1,...,xk} ∼= H|{y1,...,yk}F „heorem The random graph G(Gn, p) follows zero-one k-law if and only if P(Duplicator wins the game EHR(G(Gn, p), G(Gm, p), k)) → 1 as n, m → ∞. 10/22
  • 19. pull level extension property he(nitionF „he gr—ph G = (V, E) is s—id to s—tisfy full level t extension property if for —ny verti™es v1, . . . , vl, u1, . . . , ur (l + r ≤ t) there exists — vertex v —dj—™ent to v1, . . . , vl —nd nonE—dj—™ent to u1, . . . , urF 11/22
  • 20. pull level extension property he(nitionF „he gr—ph G = (V, E) is s—id to s—tisfy full level t extension property if for —ny verti™es v1, . . . , vl, u1, . . . , ur (l + r ≤ t) there exists — vertex v —dj—™ent to v1, . . . , vl —nd nonE—dj—™ent to u1, . . . , urF €roposition Let G(Gn, p) satisfy full level (k − 1) extension property asymptotically almost surely. Then the random graph G(Gn, p) follows zero-one k-law. 11/22
  • 21. pull level extension property he(nitionF „he gr—ph G = (V, E) is s—id to s—tisfy full level t extension property if for —ny verti™es v1, . . . , vl, u1, . . . , ur (l + r ≤ t) there exists — vertex v —dj—™ent to v1, . . . , vl —nd nonE—dj—™ent to u1, . . . , urF €roposition Let G(Gn, p) satisfy full level (k − 1) extension property asymptotically almost surely. Then the random graph G(Gn, p) follows zero-one k-law. goroll—ry Let G(Gn, p) satisfy full level t extension property a.a.s for every t ∈ N. Then the random graph G(Gn, p) follows the zero-one law. 11/22
  • 22. pull level extension property for r—ndom dist—n™e gr—ph €roposition Let a(n) = αn, α ∈ Q, 0 < α < 1. Then G(Gdist ni , p) satises full level t extension property a.a.s for every t ∈ N if and only if c = α2n and ∀m ∈ N m|ni for suciently large i. 12/22
  • 23. pull level extension property for r—ndom dist—n™e gr—ph €roposition Let a(n) = αn, α ∈ Q, 0 < α < 1. Then G(Gdist ni , p) satises full level t extension property a.a.s for every t ∈ N if and only if c = α2n and ∀m ∈ N m|ni for suciently large i. €roposition Let a(n) = αn, c = α2n, α ∈ Q, 0 < α < 1, t ≤ 5. Then G(Gdist ni , p) satises full level t extension property a.a.s if and only if Dt|a(ni) − c(ni) for suciently large i, where D2 = 1, D3 = 2, D4 = 6, D5 = 60. 12/22
  • 24. eroEone kEl—ws for r—ndom dist—n™e gr—ph xot—tionF a = αn, c = α2n, α = s/q, (s, q) = 1. 13/22
  • 25. eroEone kEl—ws for r—ndom dist—n™e gr—ph xot—tionF a = αn, c = α2n, α = s/q, (s, q) = 1. „heorem @zeroEone 4El—wA The random graph G(Gdist n , p) follows extended zero-one 4-law. The sequence G(Gdist ni , p) follows zero-one 4-law if and only if ∃i0 such that all the numbers a(ni) − c(ni) for i > i0 have the same parity. 13/22
  • 26. eroEone kEl—ws for r—ndom dist—n™e gr—ph xot—tionF a = αn, c = α2n, α = s/q, (s, q) = 1. „heorem @zeroEone 4El—wA The random graph G(Gdist n , p) follows extended zero-one 4-law. The sequence G(Gdist ni , p) follows zero-one 4-law if and only if ∃i0 such that all the numbers a(ni) − c(ni) for i > i0 have the same parity. „heorem @zeroEone 5El—wA Let a sequence {ni} be such that a(ni) − c(ni) are even for suciently large i. Then G(Gdist ni , p) follows extended zero-one 5-law, G(Gdist ni , p) follows zero-one 5-law if and only if ∃i0 such that either ∀i > i0 3|a(ni) − c(ni) or ∀i > i0 3 a(ni) − c(ni). 13/22
  • 27. eroEone kEl—ws for r—ndom dist—n™e gr—ph „heorem @zeroEone 6El—wA Let q = 5 and a sequence {ni} be such that a(ni) − c(ni) are divisible by 12 for suciently large i. Then G(Gdist ni , p) follows extended zero-one 6-law, G(Gdist ni , p) follows zero-one 6-law if and only if ∃i0 such that either ∀i > i0 5|a(ni) − c(ni) or ∀i > i0 5 a(ni) − c(ni). 14/22
  • 28. hisproof of extended zeroEone l—ws for r—ndom dist—n™e gr—ph ∀β > 0 min(p, 1 − p)|V dist n |β → ∞ —s n → ∞. (∗) „heorem @disproof of extended zeroEone 6El—wA Let one of the following two cases take place: q = 5 and a sequence {ni} is such that a(ni) − c(ni) are not divisible by 5 for suciently large i, α = 1 2 and a sequence {ni} is such that a(ni) − c(ni) are not divisible by 4 for suciently large i. Then there exists a function p(n) satisfying (∗) such that G(Gdist ni , p) doesn't follow extended zero-one 6-law. 15/22
  • 29. hisproof of extended zeroEone l—ws for r—ndom dist—n™e gr—ph „heorem @disproof of extended zeroEone l—wA Let q be even, α ∈ (1 4, 3 4) and a sequence {ni} be such that a(ni) − c(ni) are not divisible by 4 for suciently large i. Then there exists a function p(n) satisfying (∗) such that G(Gdist ni , p) doesn't follow extended zero-one law. 16/22
  • 30. ƒpe™i—l sets of verti™es he(nitionF †erti™es v1, . . . , vt of — gr—ph G = (V, E) —re s—id to form — spe™i—l tEset if there doesn9t exist — vertex v ∈ V —dj—™ent to —ll of the verti™es v1, . . . , vtF 17/22
  • 31. ƒpe™i—l sets of verti™es he(nitionF †erti™es v1, . . . , vt of — gr—ph G = (V, E) —re s—id to form — spe™i—l tEset if there doesn9t exist — vertex v ∈ V —dj—™ent to —ll of the verti™es v1, . . . , vtF vet Rt ˜e — property of sp—nning su˜gr—phs of GnX for —ny verti™es v1, . . . , vt not forming — spe™i—l tEset in Gn —nd for —ny su˜set U ⊆ {v1, . . . , vt} there exists — vertex v —dj—™ent to —ll verti™es from U —nd nonE—dj—™ent to —ll verti™es from {v1, . . . , vt} UF 17/22
  • 32. ƒpe™i—l sets of verti™es he(nitionF †erti™es v1, . . . , vt of — gr—ph G = (V, E) —re s—id to form — spe™i—l tEset if there doesn9t exist — vertex v ∈ V —dj—™ent to —ll of the verti™es v1, . . . , vtF vet Rt ˜e — property of sp—nning su˜gr—phs of GnX for —ny verti™es v1, . . . , vt not forming — spe™i—l tEset in Gn —nd for —ny su˜set U ⊆ {v1, . . . , vt} there exists — vertex v —dj—™ent to —ll verti™es from U —nd nonE—dj—™ent to —ll verti™es from {v1, . . . , vt} UF €roposition For every t ∈ N the random graph G(Gdist n , p) satisfyes Rt a.a.s. 17/22
  • 33. €roof of zeroEone kEl—wsX spe™i—l sets of verti™es without edges ƒuppose @IA Gn = (Vn, En) doesn9t h—ve spe™i—l (t − 1)Esets @PA G(Gn, p) s—tisfyes Rt —F—FsF 18/22
  • 34. €roof of zeroEone kEl—wsX spe™i—l sets of verti™es without edges ƒuppose @IA Gn = (Vn, En) doesn9t h—ve spe™i—l (t − 1)Esets @PA G(Gn, p) s—tisfyes Rt —F—FsF €roposition Let a sequence Gn = (Vn, En) satisfy (1), (2) and the following conditions: Gn has special t-sets, for every special t-set any two of its vertices are non-adjacent. Then the random graph G(Gn, p) follows zero-one (t + 1)-law. 18/22
  • 35. €roof of zeroEone kEl—wsX spe™i—l sets of verti™es with edges €roposition Suppose Gn = (Vn, En) satises (1), (2) and for any vertices v1, . . . , vi where i < t one of the following holds: for any vertex vi+1 such that v1, . . . , vi+1 can be extended to a special t-set there exist Ω(|Vn|β) dierent vertices each of which can be mapped onto vi+1 by an automorphism of Gn xing v1, . . . , vi (where β is a positive constant), |{(vi+1, . . . , vt) : {v1, . . . , vt} is a special t-set}| = O(1). Then the random graph G(Gn, p) follows extended zero-one (t + 1)-law. 19/22
  • 36. hisproof of extended zeroEone kEl—ws vet L ˜e — property of su˜gr—phs G ⊆ GnX for —ny (v1, . . . , vi) th—t ™—n ˜e extended to — spe™i—l tEset with edges in Gn there exist vi+1, . . . , vt extending (v1, . . . , vi) to — spe™i—l tEset with edges in GF 20/22
  • 37. hisproof of extended zeroEone kEl—ws vet L ˜e — property of su˜gr—phs G ⊆ GnX for —ny (v1, . . . , vi) th—t ™—n ˜e extended to — spe™i—l tEset with edges in Gn there exist vi+1, . . . , vt extending (v1, . . . , vi) to — spe™i—l tEset with edges in GF vet K(v1, . . . , vi) ˜e the num˜er of (vi+1, . . . , vt) extending (v1, . . . , vi) to — spe™i—l tEset with edges in GnF 20/22
  • 38. hisproof of extended zeroEone kEl—ws vet L ˜e — property of su˜gr—phs G ⊆ GnX for —ny (v1, . . . , vi) th—t ™—n ˜e extended to — spe™i—l tEset with edges in Gn there exist vi+1, . . . , vt extending (v1, . . . , vi) to — spe™i—l tEset with edges in GF vet K(v1, . . . , vi) ˜e the num˜er of (vi+1, . . . , vt) extending (v1, . . . , vi) to — spe™i—l tEset with edges in GnF sf there exists (v1, . . . , vi) with K(v1, . . . , vi) → ∞, K(v1, . . . , vi) = |Vn|o(1) , then PGn,p(L) ™—n —ppro—™h —ny num˜er from (0, 1)F 20/22
  • 39. hisproof of extended zeroEone kEl—ws ‚epl—™e L ˜y — (rstEorder property LX L = ∀v1 . . . ∀vi ∃vi+1 . . . ∃vt Q(v1, . . . , vt), where Q —pproxim—tely s—ys th—t either (v1, . . . , vi) ™—n9t ˜e extended to — spe™i—l tEset with edges in Gn or (v1, . . . , vt) forms — spe™i—l tEset with edges in G(Gn, p)F 21/22
  • 40. „h—nks Thank you for your attention! 22/22