## 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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## References

- [1]
N. Alon and J. Spencer:

*The Probabilistic Method*, 3rd ed., Wiley, New York, 2007. - [2]
B. Bollobás and V. Nikiforov: Books in graphs,

*European J. Combin.***26**(2005), 259–270. - [3]
F. Chung and R. Graham:

*Erdős on graphs. His legacy of unsolved problems*, A K Peters, Ltd., Wellesley, MA, 1998. - [4]
C.S. Edwards: A lower bound for the largest number of triangles with a common edge, 1977 (unpublished manuscript).

- [5]
P. Erdős: On a theorem of Rademacher-Turán,

*Illinois J. Math.***6**(1962), 122–127. - [6]
P. Erdős: Some problems on finite and infinite graphs,

*Logic and combinatorics*(Arcata, Calif., 1985), 223–228,*Contemp. Math.***65**, Amer. Math. Soc., Providence, RI, 1987. - [7]
P. Erdős: Problems and results in combinatorial analysis and graph theory, In:

*Proceedings of the First Japan Conference on Graph Theory and Applications*(Hakone, 1986),*Discrete Math.***72**(1988), 81–92. - [8]
P. Erdős: Some of my favourite problems in various branches of combinatorics,

*Combinatorics***92**(Catania, 1992),*Matematiche*(Catania)**47**(1992), 231–240 (1993). - [9]
P. Erdős, R. Faudree and E. Györi: On the book size of graphs with large minimum degree,

*Studia Sci. Math. Hungar.***30**(1995), 25–46. - [10]
P. Erdős, R. Faudree, C. Rousseau: Extremal problems and generalized degrees,

*Graph Theory and Applications*(Hakone, 1990),*Discrete Math.***127**(1994), 139–152. - [11]
R. J. Faudree, C. C. Rousseau, J. Sheehan: More from the good book, In:

*Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and Computing, Florida Atlantic Univ., Boca Raton, Fla.*, 1978, 289–299. Congress. Numer., XXI, Utilitas Math., Winnipeg, Man., 1978. - [12]
J. Fox: A new proof of the graph removal lemma,

*Annals of Mathematics***174**(2011), 561–579. - [13]
N. Khadžiivanov and V. Nikiforov: Solution of a problem of P. Erdős about the maximum number of triangles with a common edge in a graph,

*C. R. Acad. Bulgare Sci.***32**(1979) 1315–1318. - [14]
V. Nikiforov and C. C. Rousseau: Large generalized books are

*p*-good,*J. Combin. Theory Ser. B***92**(2004), 85–97. - [15]
V. Nikiforov and C. C. Rousseau: A note on Ramsey numbers for books,

*J. Graph Theory***49**(2005), 168–176. - [16]
V. Nikiforov and C. C. Rousseau: Book Ramsey numbers, I.

*Random Structures Algorithms***27**(2005), 379–400. - [17]
V. Nikiforov, C. C. Rousseau, and R. H. Schelp: Book Ramsey numbers and quasi-randomness,

*Combin. Probab. Comput.***14**(2005), 851–860. - [18]
C. C. Rousseau and J. Sheehan: On Ramsey numbers for books,

*J. Graph Theory***2**(1978), 77–87. - [19]
I. Z. Ruzsa and E. Szemerédi:

*Triple systems with no six points carrying three triangles. Combinatorics*(Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, 939–945, Colloq. Math. Soc. János Bolyai, 18, North-Holland, Amsterdam-New York, 1978. - [20]
B. Sudakov: Large

*K*_{r}-free subgraphs in*K*_{s}-free graphs and some other Ramseytype problems,*Random Structures Algorithms***26**(2005), 253–265.

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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