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}
}
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 = ∞. 

Topics from this paper

On the Number of Graphs Without Large Cliques
TLDR
The proof is based on the recent hypergraph container theorems of Saxton and Thomason and Balogh, Morris, and Samotij, in combination with a theorem of Lovasz and Simonovits. Expand
Perfect Graphs of Fixed Density: Counting and Homogeneous Sets
TLDR
It is shown that almost all graphs in n(c) have homogeneous sets of linear size, which answers a question raised by Loebl and co-workers. Expand
Supersaturated sparse graphs and hypergraphs
A central problem in extremal graph theory is to estimate, for a given graph $H$, the number of $H$-free graphs on a given set of $n$ vertices. In the case when $H$ is not bipartite, fairly preciseExpand
Excluding Induced Subgraphs III: A General Asymptotic
In this article we study asymptotic properties of the class of graphs not containing a fixed graph H as an induced subgraph. In particular we show that the number Forbn★(H) of such graphs on nExpand
Universally Sparse Hypergraphs with Applications to Coding Theory
TLDR
The new lower bound provides improved constructions for several seemingly unrelated topics in Coding Theory, namely, Parent-Identifying Set Systems, uniform Combinatorial Batch Codes and optimal Locally Recoverable Codes. Expand
The number of hypergraphs and colored Hypergraphs with hereditary properties
As an application of Szemeredi's regularity lemma, Erdos-Frankl-Rodl (1986) showed that the number of graphs on vertex set {1,2,...n} with a monotone class P is $2^{(1+o(1))ex(n,P)n^2/2}$ whereExpand
On An Extremal Hypergraph Problem Of Brown, Erdős And Sós
TLDR
This work partially resolve a problem raised by Brown, Erdős and Sós in 1973, by showing that for any fixed 2≤k<r, n - o ( 1,3) = o ( n k) in an r-uniform hypergraph on n vertices. Expand
Counting $r$-graphs without forbidden configurations
One of the major problems in combinatorics is to determine the number of r-uniform hypergraphs (r-graphs) on n vertices which are free of certain forbidden structures. This problem dates back to theExpand
The number of graphs with large forbidden subgraphs
TLDR
It is shown that if H"1,H"2,... are graphs with |H"n|=o(logn) and @g(H" n)=r"n+1, then the number S"n of graphs of order n not containing H"n satisfies log"2S"n=(1-1/r" n+o(1))(n2). Expand
On 4-uniform hypergraphs without 3 edges on 6 vertices
Let $f_r(n)$ be the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices not containing a subgraph with $r+2$ vertices and $3$ edges. It is known that $f_2(n) = \left\lfloorExpand
...
1
2
3
4
5
...

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
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
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
...
1
2
3
4
...