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| Paper IPM / M / 14668 |
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| Abstract: | |||||
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Recently, determining the Ramsey numbers of loose paths and cycles in uniform hypergraphs has received considerable attention. It has been shown that the 2-color Ramsey number of a k-uniform loose cycle Ckn, R(Ckn,Ckn), is asymptotically 12(2k�??1)n. Here we conjecture that for any n�?�m�?�3 and k�?�3,
R(Pkn,Pkm)=R(Pkn,Ckm)=R(Ckn,Ckm)+1=(k�??1)n+�??m+12�??.
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 m�?�3 and k�?�4 the conjecture is equivalent to (only) the last equality for any 2m�?�n�?�m�?�3. Consequently, the proof for m=3 follows.
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