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Paper   IPM / M / 15575
School of Mathematics
  Title:   Ramsey numbers of 4-uniform loose cycles
1.  Gholamreza Omidi
2.  Maryam Shahsiah
  Status:   Published
  Journal: Discrete Appl. Math.
  Vol.:  230
  Year:  2017
  Pages:   112-120
  Supported by:  IPM
Gyárfás, Sárközy and Szemerédi proved that the 2-color Ramsey number R(Cnk,Cnk) of a k-uniform loose cycle Cnk is asymptotically 12(2k−1)n, generating the same result for k=3 due to Haxell et al. Concerning their results, it is conjectured that for every n≥m≥3 and k≥3, R(Cnk,Cmk)=(k−1)n+⌊m−12⌋. In 2014, the case k=3 is proved by the authors. Recently, the authors showed that this conjecture is true for n=m≥2 and k≥8. Their method can be used for case n=m≥2 and k=7, but more details are required. The only open cases for the above conjecture when n=m are k=4,5,6. Here, we investigate the case k=4, and we show that the conjecture holds for k=4 when n>m or n=m is odd. When n=m is even, we show that R(Cn4,Cn4) is between two values with difference one.

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