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Paper IPM / M / 15575  


Abstract:  
GyÃ¡rfÃ¡s, SÃ¡rkÃ¶zy and SzemerÃ©di proved that the 2color Ramsey number R(Cnk,Cnk) of a kuniform loose cycle Cnk is asymptotically 12(2kâ1)n, generating the same result for k=3 due to Haxell et al. Concerning their results, it is conjectured that for every nâ¥mâ¥3 and kâ¥3, R(Cnk,Cmk)=(kâ1)n+âmâ12â. In 2014, the case k=3 is proved by the authors. Recently, the authors showed that this conjecture is true for n=mâ¥2 and kâ¥8. Their method can be used for case n=mâ¥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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