Maximal trades (Q2761060)

Volume \(\text{vol}(T)\) of a \(t\)-\((v,k)\) trade \(T=(T^+,T^-)\) means the common size of \(T^+\) and \(T^-\). The trade \(T\) is called maximal, if its volume is maximal with respect to \(t,v,k\). The authors determine the volume of maximal 2-\((v,3)\) trades for \(v>5\) and \(v\not \equiv 5\bmod 6\) (for example, one gets \(4m(4m+1)(m-1)/3\) if \(v\) is divisible by 4). If \(v=6m+5\), then the volume is \(\leq 18m^3+33m^2+19m+2\), and this result is known to be optimal for \(m=1,2\).