Efficient Testing of Hypergraphs
We investigate a basic problem in combinatorial property testing, in the sense of Goldreich, Goldwasser, and Ron [9,10], in the context of 3-uniform hypergraphs, or 3-graphs for short. As customary, a 3-graph F is simply a collection of 3-element sets. Let Forbind(n, F) be the family of all 3-graphs on n vertices that contain no copy of F as an induced subhypergraph. We show that the property “H ∈ Forbind(n, F)” is testable, for any 3-graph F. In fact, this is a consequence of a new, basic combinatorial lemma, which extends to 3-graphs a result for graphs due to Alon, Fischer, Krivelevich, and Szegedy [2,3].
Indeed, we prove that if more than ξn 3 (V > 0) triples must be added or deleted from a 3-graph H on n vertices to destroy all induced copies of F, then H must contain ≥ cn |V(F)| induced copies of F, as long as n ≥ n 0(ξ, F). Our approach is inspired in [2,3], but the main ingredients are recent hypergraph regularity lemmas and counting lemmas for 3-graphs.
KeywordsBipartite Graph Triple System Arithmetic Progression Property Testing Large Graph
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