## On the structure of edge graphs

Venue: | Bull . London Math . Soc |

Citations: | 18 - 3 self |

### BibTeX

@ARTICLE{Bollobas_onthe,

author = {Béla Bollobas and Paul Erdős},

title = {On the structure of edge graphs},

journal = {Bull . London Math . Soc},

year = {}

}

### OpenURL

### Abstract

Every graph appearing in this note is a finite edge graph without loops and multiple edges. Denote by G(n, m) a graph with n vertices and ni edges. K r (t) denotes a graph with r groups of t vertices each, in which two vertices are connected if and only if they belong to different groups. By dividing n vertices into r-1 almost equal groups and connecting the points in different groups one obtains a graph on n vertices with ((r- 2)/2(r- 1) + u (1)) 11 2 edges which does not contain a Kr (l). On the other hand, it was shown by Erdős and Stone [7] that ((r- 2)/2(r- 1) + s) n2 (e> 0) edges assure already the existence of a Kr (t), where t- ~ oo as n-p co. This result is the inost essential part of the theorems on the structure of extremal graphs, see e.g. [3], [4], [6], [9]. Let us formulate the result of Erdős and Stone more precisely. Given n, r and e, put rn = [((r-2)/2(r-1)+e)n2] ([x] denotes the integer part of x) and define g(n, r, e) = min {t: every G(n, in) contains a K r (t)}. Erdős and Stone proved that if n is large enough then (1r _ i (n)) ' < g(n, r, c), where i s denotes the s times iterated logarithm. They also stated that for any fixed S> 0 and large enough n the same method gives (lr- 1(n)) '-ó C g(71, 1', e). In [7] Erdős and Stone also expected that l r _ 1 (n) is, in fact, the proper order of g(n, r, e) if e is small enough. For r = 2 this was stated in [1]. In [2] Erdős announced that given e> 0 and r> 2 there exists a constant c '> 0 such that c ' (log n) 1 l ( r-1) < g(n, r, c), and thought that g(n, r, a) will turn out to be of order (log n) 1 1 ( r-1) The aim of this note is to show that for r> 2 the situation is rather different from what seeined likely. The two theorems we prove (of which the second is an easy exercise in the vein of [5]) show that for any r and 0 < e < 112(r-1) there are constants c, and c 2> 0 such that CI log 11 g01, r, e) s c2 log n if n is sufficiently large and c 2-> 0 as a--, 0. The following lemma is needed in the proof of Theorem 1.

### Citations

1842 | On the evolution of random graphs
- Erdős, Rényi
- 1960
(Show Context)
Citation Context ... 1( r -1) The aim of this note is to show that for r > 2 the situation is rather different from what seeined likely . The two theorems we prove (of which the second is an easy exercise in the vein of =-=[5]-=-) show that for any r and 0 < e < 112(r -1) there are constants c, and c 2 > 0 such that CI log 11 g01, r, e) s c2 log n if n is sufficiently large and c 2 -> 0 as a --, 0 . The following lemma is nee... |

63 | A limit theorem in graph theory
- Erdős, Simonovits
- 1966
(Show Context)
Citation Context ... n2 (e > 0) edges assure already the existence of a Kr (t), where t -~ oo as n -p co . This result is the inost essential part of the theorems on the structure of extremal graphs, see e .g. [3], [4], =-=[6]-=-, [9] . Let us formulate the result of Erdős and Stone more precisely . Given n, r and e, put rn = [((r-2)/2(r-1)+e)n2] ([x] denotes the integer part of x) and define g(n, r, e) = min {t : every G(n, ... |

5 |
A method for solving extremal problems in graph theory . Stability problems
- unknown authors
- 1966
(Show Context)
Citation Context ...e > 0) edges assure already the existence of a Kr (t), where t -~ oo as n -p co . This result is the inost essential part of the theorems on the structure of extremal graphs, see e .g. [3], [4], [6], =-=[9]-=- . Let us formulate the result of Erdős and Stone more precisely . Given n, r and e, put rn = [((r-2)/2(r-1)+e)n2] ([x] denotes the integer part of x) and define g(n, r, e) = min {t : every G(n, in) c... |

1 | On extremal problems of graphs and generalised graphs - unknown authors - 1965 |

1 |
some new inequalities concerning extremal properties of graphs ", Theory of graph, ed
- Of
- 1968
(Show Context)
Citation Context ... - 1) + s) n2 (e > 0) edges assure already the existence of a Kr (t), where t -~ oo as n -p co . This result is the inost essential part of the theorems on the structure of extremal graphs, see e .g. =-=[3]-=-, [4], [6], [9] . Let us formulate the result of Erdős and Stone more precisely . Given n, r and e, put rn = [((r-2)/2(r-1)+e)n2] ([x] denotes the integer part of x) and define g(n, r, e) = min {t : e... |

1 |
On the structure of linear graphs
- unknown authors
(Show Context)
Citation Context ...ecting the points in different groups one obtains a graph on n vertices with ((r - 2)/2(r - 1) + u (1)) 11 2 edges which does not contain a Kr (l) . On the other hand, it was shown by Erdős and Stone =-=[7]-=- that ((r - 2)/2(r - 1) + s) n2 (e > 0) edges assure already the existence of a Kr (t), where t -~ oo as n -p co . This result is the inost essential part of the theorems on the structure of extremal ... |

1 |
On a problem of Zarankievicz
- Sós, P
- 1954
(Show Context)
Citation Context ... 2) . For put j = j(n) _ [k log n] and suppose that there are arbitrary large values of 11 for which there is a graph G(n, m) without a K20), where m > En2 . Then by a result of Kővári, Sós and Turán =-=[8]-=- (improved by Znám [10]), En2 <i(j-1)'1jn2-('lj)+iin . That is, after dividing by In 2-('1j), we obtain that for any q > 1 there is an arbitrary large n, such that 2 sn'li -< q, which contradicts the ... |