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 ?n3 (? > 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 ? n0(?,F). Our approach is inspired in [2,3], but the main ingredients are recent hypergraph regularity lemmas and counting lemmas for 3-graphs.

[1]  W. T. Gowers,et al.  A New Proof of Szemerédi's Theorem for Arithmetic Progressions of Length Four , 1998 .

[2]  E. Szemerédi.  On sets of integers containing no four elements in arithmetic progression , 1969 .

[3]  Vojtech Rödl,et al.  The Algorithmic Aspects of the Regularity Lemma , 1994, J. Algorithms.

[4]  V. Rödl,et al.  Extremal problems on set systems , 2002 .

[5]  Dana Ron,et al.  Property testing and its connection to learning and approximation , 1998, JACM.

[6]  K. F. Roth.  On Certain Sets of Integers , 1953 .

[7]  H. Furstenberg.  Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions , 1977 .

[8]  Noga Alon,et al.  Testing subgraphs in large graphs , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[9]  Jean Bourgain,et al.  On Triples in Arithmetic Progression , 1999 .

[10]  Vojtech Rödl,et al.  Extremal problems on set systems , 2002, Random Struct. Algorithms.

[11]  Artur Czumaj,et al.  Testing Hypergraph Coloring , 2001, ICALP.

[12]  Vojtech Rödl,et al.  Regularity properties for triple systems , 2003, Random Struct. Algorithms.

[13]  E. Szemerédi.  On sets of integers containing k elements in arithmetic progression , 1975 .

[14]  K. F. Roth.  Irregularities of sequences relative to arithmetic progressions , 1967 .

[15]  K. F. Roth.  Irregularities of sequences relative to arithmetic progressions, IV , 1972 .

[16]  W. T. Gowers,et al.  A new proof of Szemerédi's theorem , 2001 .

[17]  Noga Alon,et al.  Efficient Testing of Large Graphs , 2000, Comb..