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Paper IPM / M / 14668  


Abstract:  
Recently, determining the Ramsey numbers of loose paths and cycles in uniform hypergraphs has received considerable attention. It has been shown that the 2color Ramsey number of a kuniform loose cycle Ckn, R(Ckn,Ckn), is asymptotically 12(2kï¿½??1)n. Here we conjecture that for any nï¿½?ï¿½mï¿½?ï¿½3 and kï¿½?ï¿½3,
R(Pkn,Pkm)=R(Pkn,Ckm)=R(Ckn,Ckm)+1=(kï¿½??1)n+ï¿½??m+12ï¿½??.
Recently the case k=3 is proved by the authors. In this paper, first we show that this conjecture is true for k=3 with a much shorter proof. Then, we show that for fixed mï¿½?ï¿½3 and kï¿½?ï¿½4 the conjecture is equivalent to (only) the last equality for any 2mï¿½?ï¿½nï¿½?ï¿½mï¿½?ï¿½3. Consequently, the proof for m=3 follows.
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