# Quasi-randomness and the distribution of copies of a fixed graph

@article{Shapira2008QuasirandomnessAT, title={Quasi-randomness and the distribution of copies of a fixed graph}, author={Asaf Shapira}, journal={Combinatorica}, year={2008}, volume={28}, pages={735-745} }

We show that if a graph G has the property that all subsets of vertices of size n/4 contain the “correct” number of triangles one would expect to find in a random graph G(n, 1/2), then G behaves like a random graph, that is, it is quasi-random in the sense of Chung, Graham, and Wilson [6]. This answers positively an open problem of Simonovits and Sós [10], who showed that in order to deduce that G is quasi-random one needs to assume that all sets of vertices have the correct number of triangles…

## 28 Citations

Quasi-randomness is determined by the distribution of copies of a fixed graph in equicardinal large sets

- MathematicsComb.
- 2010

It is shown that if a graph G has the property that all subsets of size αn contain the “correct” number of copies of H one would expect to find in the random graph G(n,p), then G behaves like the randomgraph G( n,p); that is, it is p-quasi-random in the sense of Chung, Graham, and Wilson [4].

The effect of induced subgraphs on quasi-randomness

- Mathematics, Computer Science
- 2010

Having the correct distribution of induced copies of any single graph H is enough to guarantee that a graph has the properties of a random one, and the proof techniques developed here, which combine probabilistic, algebraic, and combinatorial tools, may be of independent interest to the study of quasi-random structures.

More on quasi-random graphs, subgraph counts and graph limits

- MathematicsEur. J. Comb.
- 2015

Hereditary quasirandom properties of hypergraphs

- MathematicsComb.
- 2011

A natural extension of the result of Simonovits and Sós to k-uniform hypergraphs is given, answering a question of Conlon et al. and yields an alternative, and perhaps simpler, proof of one of their theorems.

Hereditary quasirandomness without regularity

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2017

Abstract A result of Simonovits and Sós states that for any fixed graph H and any ε > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every S ⊆ V(G) contains pe(H)…

Extremal Results in Random Graphs

- Mathematics
- 2013

According to Paul Erdős au][Some notes on Turan’s mathematical work, J. Approx. Theory 29 (1980), page 4]_it was Paul Turan who “created the area of extremal problems in graph theory”. However,…

Eigenvalues and Quasirandom Hypergraphs

- Mathematics
- 2013

Let p(k) denote the partition function of k. For each k 2, we describe a list of p(k) 1 quasirandom properties that a k-uniform hypergraph can have. Our work connects previous notions on hypergraph…

Extremal hypergraph theory and algorithmic regularity lemma for sparse graphs

- Mathematics, Computer Science
- 2011

This thesis provides a polynomial time algorithm which computes a regular partition for given graphs without too dense induced subgraphs and derives (asymptotically sharp) bounds on minimum degrees of uniform hypergraphs which guarantee the appearance of perfect and nearly perfect matchings.

Sigma-Algebras for Quasirandom Hypergraphs

- Mathematics
- 2013

We examine the correspondence between the various notions of quasirandomness for k-uniform hypergraphs and sigma-algebras related to measurable hypergraphs. This gives a uniform formulation of most…

EIGENVALUES AND LINEAR QUASIRANDOM HYPERGRAPHS

- MathematicsForum of Mathematics, Sigma
- 2015

Let $p(k)$ denote the partition function of $k$. For each $k\geqslant 2$, we describe a list of $p(k)-1$ quasirandom properties that a $k$-uniform hypergraph can have. Our work connects previous…

## References

SHOWING 1-10 OF 21 REFERENCES

Hereditarily extended properties, quasi-random graphs and not necessarily induced subgraphs

- MathematicsComb.
- 1997

Here propertiesP which do not imply quasi-randomnes for sequences of graphs on their own, but do imply if they hold not only for the whole graph but also for every sufficiently large subgraph of graphs, are investigated.

Quasi-random tournaments

- MathematicsJ. Graph Theory
- 1991

A large class of tournament properties, all of which are shared by almost all random tournaments, are introduced, which have the property that tournaments possessing any one of the properties must of necessity possess them all.

Pseudo-random Graphs

- Computer Science
- 2006

Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake,…

A Diagonal Form for the Incidence Matrices of t-Subsets vs.k-Subsets

- MathematicsEur. J. Comb.
- 1990

A certain class of incidence matrices

- Mathematics
- 1966

1. Let S be a set consisting of K elements, and call any subset of S containing precisely m elements an m-set [2]. We wish to study incidence matrices obtained in the following manner: Let K > mrn_ n…

Quasirandom Groups

- MathematicsCombinatorics, Probability and Computing
- 2008

It is shown that there exists a constant c > 0 such that every finite group G has a product-free subset of size at least c|G|: that is, a subset X that does not contain three elements x, y and z with xy = z.

Quasi-Random Hypergraphs

- MathematicsRandom Struct. Algorithms
- 1990

A large equivalence class of properties shared by most hypergraphs, including so-called random hyper graphs, are described, which shows that many global properties of hyperGraphs are actually consequences of simple local conditions.

A Note on the Ranks of Set-Inclusion Matrices

- MathematicsElectron. J. Comb.
- 2001

A recurrence relation is derived for the rank (over most fields) of the set-inclusion matrices on a finite ground set.