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2014

Limits, Regularity and Removal for Finite Structures

Aroskar, Ashwini, Cummings, James

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Our work builds on known results for k-uniform hypergraphs including the existence of limits, a Regularity Lemma and a Removal Lemma.

Our main tool here is a theory of measures on ultraproduct spaces which establishes a correspondence between ultraproduct spaces and Euclidean spaces.
First we show the existence of a limit object for convergent sequences of relational structures and as a special case, we retrieve the known limits for graphs and digraphs.
Then we extend this notion to finite models of a fixed universal theory.

Limits, Regularity and Removal for Finite Structures · Around

Built on

On Homogeneous Measure Algebras
D. Maharam · 1942
Earlier work this paper cites.
Measurable cardinals and constructible sets
D. Scott · 1961
Earlier work this paper cites.
Conversion from Nonstandard to Standard Measure Spaces and Applications in Probability Theory
P. Loeb · 1975
Earlier work this paper cites.

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Limits of Dense Graph Sequences
L. Lovász and B. Szegedy · 2006
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Flag Algebras
A. Razborov · 2007
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Extremal Problems on the Hypercube
D. Offner · 2009
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Then

Poset Limits and Exchangeable Random Posets
S. Janson · 2011
Later among the works it cites.
A Measure-theory Approach to the Theory of Dense Hypergraphs
G. Elek and B. Szegedy · 2012
Later among the works it cites.
Large Networks and Graph Limits
L. Lovász · 2012
Later among the works it cites.

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