Graphs without four‐cycles

Clapham, C. R. J.; Flockhart, A.; Sheehan, J.

doi:10.1002/jgt.3190130107

Cited by 57 publications

(54 citation statements)

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“…Here we know that ex(n, C 4 ) = (1/2 + o(1)) n 3/2 : the lower bound was proved by Erdős and Rényi [8] (and independently rediscovered by Brown [4]) and the upper bound by Erdős, Rényi, and Sós [9]. Moreover, we know ex(n, C 4 ) exactly when n = q 2 + q + 1 for any prime power q ≥ 16 (Füredi [11,12]) and when n ≤ 31 (McCuaig [17], Clapham, Flockhart, and Sheehan [5], Yuansheng and Rowlinson [23]). …”

Section: Introductionmentioning

confidence: 82%

A note on the Turán function of even cycles

Pikhurko

2012

Proc. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 82%

A note on the Turán function of even cycles

Pikhurko

2012

Proc. Amer. Math. Soc.

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“…Since K 4 is not representable in R 2 , one of the diagonals of C (1) 4 is missing inĤ n . Since K (2, 3) is not representable in R 2 , we need a path of length 3 to connect the two ends of this missing diagonal.…”

Section: N) the Equality Holds If There Is A Unit-distance Graph Witmentioning

confidence: 98%

Subdividing a Graph Toward a Unit-distance Graph in the Plane

Gervacio

Maehara

2000

European Journal of Combinatorics

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The subdivision number of a graph G is defined to be the minimum number of extra vertices inserted into the edges of G to make it isomorphic to a unit-distance graph in the plane. Let t (n) denote the maximum number of edges of a C 4 -free graph on n vertices. It is proved that the subdivision number of K n lies between n(n − 1)/2 − t (n) and (n − 2)(n − 3)/2 + 2, and that of K (m, n) equals (m − 1)(n − m) for n ≥ m(m − 1).

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“…Thus, it has dimension -1 and can be substituted for an edge in the graphs we originally constructed. After we had constructed this graph, Carol Wood pointed out that it had arisen in another context [3]; the properties we need are verified in more detail there.…”

Section: The Specific Constructionmentioning

confidence: 99%

An Almost Strongly Minimal Non-Desarguesian Projective Plane

Baldwin¹

1994

Transactions of the American Mathematical Society

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Abstract.There is an almost strongly minimal projective plane which is not Desarguesian.Zil'ber conjectured that every strongly minimal set is 'trivial', 'field-like', or 'module-like'. This conjecture was refuted by Hrushovski [4]. Varying his construction, we refute here two more precise versions of the conjecture.Zil'ber [8] calls a strongly minimal set M field-like if there is a pseudoplane definable in M. (A pseudoplane is an incidence structure such that each pair of lines intersect in only finitely many points and dually there are only finitely many lines passing through a pair of points.) This nomenclature would have been justified if the following conjecture were correct.Conjecture B of [8]. Every uncountably categorical pseudoplane is definable in an algebraically closed field and the field is definable in the pseudoplane.This conjecture has several aspects. In a model theoretic vein it limits the variety of uncountably categorical pseudoplanes. In particular, it asserts that there are only countably many such (as each must be defined in an algebraically closed field). This aspect of the conjecture is already refuted by [4]. (Hrushovski shows there are 2N° strongly minimal sets which are not locally modular. By Zil'ber's trichotomy theorem they are thus field-like.)A more geometric aspect of the problems is phrased in another Zil'ber conjecture.Conjecture C of [8]. Every uncountably categorical affine plane is Desarguesian and hence is an affine plane over an algebrically closed field.We show that the projective plane constructed here does not interpret a group and thus cannot be Desarguesian. The affine plane associated with this projective plane also fails to be Desarguesian so Conjecture C is refuted.Finally there is a more algebraic geometric conjecture behind the part of Conjecture B asserting every field-like structure is definable in a field.Received by the editors October 23, 1989 and, in revised form, May 14, 1992. 1991 Mathematics Subject Classification. Primary 03C60; Secondary 51C10. Partially supported by NSF grant 8602558.In the time since this paper was submitted (October 1989) various authors, including Herwig, Poizat, and Wagner, have suggested alternative methods to organize the proof that the generic model is (o-saturated and that no infinite group is interpretable in a structure constructed by Hrushovski's method.

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Graphs without four‐cycles

Abstract: We investigate the values of t(n), the maximum number of edges in a graph with n vertices and not containing a four-cycle. Techniques for finding these are developed and the values of t(n) for all n up to 21 are obtained. All the corresponding extremal graphs are found.

Cited by 57 publications

References 2 publications

A note on the Turán function of even cycles

A note on the Turán function of even cycles

Subdividing a Graph Toward a Unit-distance Graph in the Plane

An Almost Strongly Minimal Non-Desarguesian Projective Plane

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