The logic of random regular graphs

Haber, Simi; Krivelevich, Michael

doi:10.4310/joc.2010.v1.n4.a3

Cited by 11 publications

(12 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, so this generalizes the 0-1 law proved in [30] for Forb({K l }), l ≥ 3. More results on logical limit laws for various families of graphs appear in [25,26,32,37]. However, these results do not apply to M r , as elements of M r are not graphs.…”

Section: Introductionmentioning

confidence: 99%

Discrete Metric Spaces: Structure, Enumeration, and 0-1 Laws

Mubayi¹,

Terry²

2019

J. symb. log.

View full text Add to dashboard Cite

Fix an integer r ≥ 3. We consider metric spaces on n points such that the distance between any two points lies in {1, . . . , r}. Our main result describes their approximate structure for large n. As a consequence, we show that the number of these metric spaces isRelated results in the continuous setting have recently been proved by Kozma, Meyerovitch, Peled, and Samotij [33]. When r is even, our structural characterization is more precise, and implies that almost all such metric spaces have all distances at least r/2. As an easy consequence, when r is even we improve the error term above from o(n 2 ) to o(1), and also show a labeled first-order 0-1 law in the language L r , consisting of r binary relations, one for each element of [r]. In particular, we show the almost sure theory T is the theory of the Fraïssé limit of the class of all finite simple complete edge-colored graphs with edge colors in {r/2, . . . , r}.Our work can be viewed as an extension of a long line of research in extremal combinatorics to the colored setting, as well as an addition to the collection of known structures that admit logical 0-1 laws.

show abstract

Section: Introductionmentioning

confidence: 99%

Discrete Metric Spaces: Structure, Enumeration, and 0-1 Laws

Mubayi¹,

Terry²

2019

J. symb. log.

View full text Add to dashboard Cite

show abstract

“…There are situations in which F O has arithmetization. Let α ∈ (0, 1) be rational, then the pairs (F O, G(n, p = n α )) and (F O, G n,d=n 1−α ) have arithmetization (where G n,d=n 1−α is the random regular graph with degree d. See [29] and [17] respectively). Also, in [20] it is showed that if d ≥ 2 is a constant integer and r is a small enough constant then the pair (F O, G(n; T d , r)) has arithmetization (where T d is the d-dimensional torus and G(n; T d , r) is the random geometric graph with distance parameter r).…”

Section: Previous Resultsmentioning

confidence: 99%

Random graphs and Lindstrom quantifiers for natural graph properties

Haber,

Shelah

2015

Preprint

Self Cite

View full text Add to dashboard Cite

We study zero-one laws for random graphs. We focus on the following question that was asked by many: Given a graph property P , is there a language of graphs able to express P while obeying the zeroone law? Our results show that on the one hand there is a (regular) language able to express connectivity and k-colorability for any constant k and still obey the zero-one law. On the other hand we show that in any (semiregular) language strong enough to express Hamiltonicity one can interpret arithmetic and thus the zero-one law fails miserably. This answers a question of Blass and Harary.

show abstract

“…Using the configuration model, Lynch [30] proved the FO-convergence law for graphs with a given degree sequence (subject to some conditions on this degree sequence), which in particular covers d-regular graphs with d fixed. For dense d-regular graphs, that is when d = Θ(n), a zero-one law was proved by Haber and Krivelevich [21], who were also able to obtain an analogue of the striking result of Shelah-Spencer mentioned above for random regular graphs.…”

Section: Introductionmentioning

confidence: 82%

Logical limit laws for minor-closed classes of graphs

Heinig

Müller

Noy

et al. 2018

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

Let G be an addable, minor-closed class of graphs. We prove that the zero-one law holds in monadic second-order logic (MSO) for the random graph drawn uniformly at random from all connected graphs in G on n vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs in G on n vertices. We also prove analogues of these results for the class of graphs embeddable on a fixed surface, provided we restrict attention to first order logic (FO). Moreover, the limiting probability that a given FO sentence is satisfied is independent of the surface S. We also prove that the closure of the set of limiting probabilities is always the finite union of at least two disjoint intervals, and that it is the same for FO and MSO. For the classes of forests and planar graphs we are able to determine the closure of the set of limiting probabilities precisely. For planar graphs it consists of exactly 108 intervals, each of the same length ≈ 5.39 · 10 −6 . Finally, we analyse examples of non-addable classes where the behaviour is quite different. For instance, the zero-one law does not hold for the random caterpillar on n vertices, even in FO.

show abstract

The logic of random regular graphs

Cited by 11 publications

References 42 publications

Discrete Metric Spaces: Structure, Enumeration, and 0-1 Laws

Discrete Metric Spaces: Structure, Enumeration, and 0-1 Laws

Random graphs and Lindstrom quantifiers for natural graph properties

Logical limit laws for minor-closed classes of graphs

Contact Info

Product

Resources

About