Lower bounds for testing triangle-freeness in Boolean functions
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
The minimum size of 3-graphs without a 4-set spanning no or exactly three edges
European Journal of Combinatorics
Testing odd-cycle-freeness in Boolean functions
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Testable and untestable classes of first-order formulae
Journal of Computer and System Sciences
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Ruzsa and Szemerédi established the triangle removal lemma by proving that: For every η0 there exists c0 so that every sufficiently large graph on n vertices, which contains at most cn 3 triangles can be made triangle free by removal of at most η $$ \left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right) $$edges. More general statements of that type regarding graphs were successively proved by several authors. In particular, Alon and Shapira obtained a generalization (which extends all the previous results of this type), where the triangle is replaced by a possibly infinite family of graphs and containment is induced. In this paper we prove the corresponding result for k-uniform hypergraphs and show that: For every family ℱ of k-uniform hypergraphs and every η0 there exist constants c 0 and C 0 such that every sufficiently large k-uniform hypergraph on n vertices, which contains at most cn νF induced copies of any hypergraph F ∈ ℱ on ν F ≤ C vertices can be changed by adding and deleting at most η $$ \left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right) $$edges in such a way that it contains no induced copy of any member of ℱ.