Ramsey numbers of uniform loose paths and cycles

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 Cnk, R(Cnk,Cnk), is asymptotically 12(2k−1)n. Here we conjecture that for any n≥m≥3 and k≥3,R(Pnk,Pmk)=R(Pnk,Cmk)=R(Cnk,Cmk)+1=(k−1)n+m+12.Recently the case k=3 was 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. Finally we give a proof for the case m=3.