Minimal functions on the random graph

@article{Bodirsky2010MinimalFO,
  title={Minimal functions on the random graph},
  author={Manuel Bodirsky and Michael Pinsker},
  journal={Israel Journal of Mathematics},
  year={2010},
  volume={200},
  pages={251-296}
}
We show that there is a system of 14 non-trivial finitary functions on the random graph with the following properties: Any non-trivial function on the random graph generates one of the functions of this system by means of composition with automorphisms and by topological closure, and the system is minimal in the sense that no subset of the system has the same property. The theorem is obtained by proving a Ramsey-type theorem for colorings of tuples in finite powers of the random graph, and by… Expand
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References

SHOWING 1-10 OF 47 REFERENCES
Reducts of Ramsey structures
TLDR
A survey of results in model theory and theoretical computer science obtained recently by the authors in this context, which approaches the problem of classifying the reducts of countably infinite ordered homogeneous Ramsey structures in a finite language, and certain decidability questions connected with such reduCTs. Expand
Schaefer's theorem for graphs
TLDR
It is proved that either Psi is contained in one out of 17 classes of graph formulas and the corresponding problem can be solved in polynomial time, or the problem is NP-complete. Expand
Constraint Satisfaction with Countable Homogeneous Templates
TLDR
It is proved that the primitive positive definable relations over an ω-categorical structure Γ are precisely the relations that are preserved by the polymorphisms of Γ. Expand
Reducts of Random Hypergraphs
  • Simon Thomas
  • Mathematics, Computer Science
  • Ann. Pure Appl. Log.
  • 1996
TLDR
It is shown that there exist only finitely many closed permutation groups G such that AUt(rk ) < G < &vn( rk) and each of the associated reducts of rk is homogeneous with respect to a finite relational language. Expand
The complexity of satisfiability problems
TLDR
An infinite class of satisfiability problems is considered which contains these two particular problems as special cases, and it is shown that every member of this class is either polynomial-time decidable or NP-complete. Expand
Oligomorphic clones
Abstract.A permutation group on a countably infinite domain is called oligomorphic if it has finitely many orbits of finitary tuples. We define a clone on a countable domain to be oligomorphic if itsExpand
The Endomorphism Monoid of the Random Graph has Uncountably Many Ideals
Abstract In this note we prove that the monoid End(R) of all endomorphisms of the random graph R is not simple. On the contrary, the lattice of ideals of End(R) embeds the poset of all subsets of ω,Expand
All countable monoids embed into the monoid of the infinite random graph
TLDR
It is proved that the full transformation monoid on a countably infinite set is isomorphic to a submonoid of End(R), the endomorphism monoid of the infinite random graph R, which has an undecidable universal theory. Expand
The 116 reducts of (Q, <, a)
This article aims to classify those reducts of expansions of (Q, <) by unary predicates which eliminate quantifiers, and in particular to show that, up to interdefinability, there are only finitelyExpand
Topological Birkhoff
One of the most fundamental mathematical contributions of Garrett Birkhoff is the HSP theorem, which implies that a finite algebra B satisfies all equations that hold in a finite algebra A of theExpand
...
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3
4
5
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