# The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent

@article{Erds1986TheAN,
title={The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent},
author={Paul Erd{\"o}s and Peter Frankl and Vojtech R{\"o}dl},
journal={Graphs and Combinatorics},
year={1986},
volume={2},
pages={113-121}
}
• Published 1986
• Computer Science, Mathematics
• Graphs and Combinatorics
AbstractLetH be a fixed graph of chromatic numberr. It is shown that the number of graphs onn vertices and not containingH as a subgraph is $$2^{(\begin{array}{*{20}c} n \\ 2 \\ \end{array} )(1 - \frac{1}{{r - 1}} + o(1))}$$ . Lethr(n) denote the maximum number of edges in anr-uniform hypergraph onn vertices and in which the union of any three edges has size greater than 3r − 3. It is shown thathr(n) =o(n2) although for every fixedc < 2 one has limn→∞hr(n)/nc = ∞.
266 Citations

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#### References

SHOWING 1-10 OF 33 REFERENCES
On the structure of linear graphs
Denote byG(n; m) a graph ofn vertices andm edges. We prove that everyG(n; [n2/4]+1) contains a circuit ofl edges for every 3 ≦l<c2n, also that everyG(n; [n2/4]+1) contains ake(un, un) withun=[c1Expand
An extremal problem in graph theory
• Mathematics
• 1970
G ( n;l ) will denote a graph of n vertices and l edges. Let f 0 ( n, k ) be the smallest integer such that there is a G ( n;f 0 (n, k )) in which for every set of k vertices there is a vertex joinedExpand
A LIMIT THEOREM IN GRAPH THEORY
• 1966
In this paper G(n ; I) will denote a graph of n vertices and l edges, K„ will denote the complete graph of p vertices G (p ; (PA and K,(p i , . . ., p,) will denote the rchromatic graph with p iExpand
On some extremal problems on r-graphs
• P. Erdös
• Computer Science, Mathematics
• Discret. Math.
• 1971
It is proved that to every e > 0 and integer t there is an n"0 = n" 0(t,@e) so that every G(^r)(n;[(c"r","l+@e)(nR)]) has lt vertices x"t(^j), l= l. Expand
Lower bounds for Turán's problem
• Mathematics, Computer Science
• Graphs Comb.
• 1985
It is proved that fora≥1 fixed andt sufficiently largeT(n, t+a,t)>(1-a(a+4+o(1))logt/(at)(tn holds). Expand
Regular Partitions of Graphs
Abstract : A crucial lemma in recent work of the author (showing that k-term arithmetic progression-free sets of integers must have density zero) stated (approximately) that any large bipartite graphExpand
On Sets of Integers Which Contain No Three Terms in Arithmetical Progression.
• F. Behrend
• Mathematics, Medicine
• Proceedings of the National Academy of Sciences of the United States of America
• 1946
By a modification of Salem and Spencer' method, the better estimate 1-_2/2log2 + e v(N) > N VloggN is shown. Expand
On universality of graphs with uniformly distributed edges
• V. Rödl
• Computer Science, Mathematics
• Discret. Math.
• 1986
Abstract We prove that sufficiently large graphs with sufficiently many ‘uniformly distributed’ edges contain all small graphs as induced subgraphs. This fails to be true for k-uniform hypergraphsExpand
Problems and Results in Combinatorial Analysis
I gave many lectures by this and similar titles, many in fact in these conferences and I hope in my lecture in 1978 I will give a survey of the old problems and describe what happened to them. In theExpand
An exact result for 3-graphs
• Computer Science, Mathematics
• Discret. Math.
• 1984
This paper proves Theorem 1 which gives a full description of families of 3-subsets in which any 4 points contain 0 or 2 members of the family. Expand