Limits, Regularity and Removal for Relational and Weighted Structures
The Szemeredi Regularity Lemma states that any graph can be well-approximated by graphs that are almost random. A well-known application of the Szemeredi Regularity Lemma is in the proof of the Removal Lemma for graphs. There are several extensions of the Regularity Lemma to hypergraphs. 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. We show the existence of a limit object for sequences of relational structures and as a special case, we retrieve the known limits for graphs and digraphs. We also state and prove a Regularity Lemma, a Removal Lemma and a Strong Removal Lemma for relational structures. The Strong Removal Lemma deals with the removal of a family of relational structures and has applications in property testing.
We have also extended the above correspondence to measurable functions on the ultraproduct and Euclidean spaces. This enabled us to find limit objects for sequences of weighted structures and these can be seen as generalizations of the limits we have obtained for relational structures. We also formulate and prove Regularity and Removal Lemmas for weighted structures.