An Explicit Construction for a Ramsey Problem

An explicit coloring of the edges of K n is constructed such that every copy of K 4 has at least four colors on its edges. As n → ∞, the number of colors used is n 1/2+o(1). This improves upon the previous bound of O(n 2/3) due to Erdős and Gyárfás obtained by probabilistic methods. The exponent 1/2 is optimal, since it is known that at least Ω(n 1/2) colors are required in such a coloring.

The coloring is related to constructions giving lower bounds for the multicolor Ramsey number r k (C 4). It is more complicated however, because of restrictions imposed on interactions between color classes.

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Correspondence to Dhruv Mubayi*.

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* Research supported in part by NSF Grant No. DMS–9970325.

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Mubayi*, D. An Explicit Construction for a Ramsey Problem. Combinatorica 24, 313–324 (2004). https://doi.org/10.1007/s00493-004-0019-6

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Mathematics Subject Classification (2000):

  • 05C35
  • 05C55
  • 05D10