On a problem of Erdos and Rothschild on edges in triangles
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Fox_On a problem.pdf
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286.72 KB
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Author(s)
Fox, Jacob
Loh, Po-Shen
Date Issued
December 2012
Journal
Combinatorica
Publisher
Springer-Verlag
Citation
Fox, Jacob, and Po-Shen Loh. “On a Problem of Erdös and Rothschild on Edges in Triangles.” Combinatorica 32.6 (December 2012), p.619-628.
Version
Author's final manuscript
Abstract
Erdos and Rothschild asked to estimate the maximum number, denoted by h(n; c), such that
every n-vertex graph with at least cn2 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, Erdos asked in 1987 to determine whether for every c > 0 there is epsilon > 0 such that h(n; c) > n epsilon for all su ciently large n. We prove that h(n; c) = nO(1= log log n) for every xed c < 1=4. This gives a negative answer to the question of Erdos, and is best possible in terms of the range for c, as it is known that every n-vertex graph with more than n2=4 edges contains an edge that is in at least n=6 triangles.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
http://dx.doi.org/10.1007/s00493-012-2844-3
https://doi.org/10.1007/s00493-012-2844-3
