On a problem of Erdös and Rothschild on edges in triangles

Abstract

Erdös and Rothschild asked to estimate the maximum number, denoted by h(n, c), such that every n-vertex graph with at least cn 2 edges, each of which is contained in at least one triangle, must contain an edge that is in at least h(n, c) triangles. In particular, Erdös asked in 1987 to determine whether for every c > 0 there is ε > 0 such that h(n,c) > n ε for all sufficiently large n. We prove that h(n,c) = n O(1/loglogn) for every fixed c < 1/4. This gives a negative answer to the question of Erdős, and is best possible in terms of the range for c, as it is known that every n-vertex graph with more than n 2/4 edges contains an edge that is in at least n/6 triangles.

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Correspondence to Jacob Fox.

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Research supported by a Simons Fellowship and NSF grant DMS-1069197.

Research supported by an NSA Young Investigators Grant and a USA-Israel BSF grant.

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Fox, J., Loh, PS. On a problem of Erdös and Rothschild on edges in triangles. Combinatorica 32, 619–628 (2012). https://doi.org/10.1007/s00493-012-2844-3

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Mathematics Subject Classification (2000)

  • 05C35