Improved LP lower bounds for difference triangle sets (Q1300349)

An \((I,J)\)-difference triangle set is a collection of \(I\) sets \(A_i= \{a_{i0},\dots, a_{ij}\}\) of \(J+1\) integers so that the integers in \(\{a_{ij}- a_{ik}: 1\leq i\leq I\) and \(0\leq k<j\leq J\}\) are all positive and distinct. Then \(M(I,J)\) denotes the minimum over all \((I,J)\)-difference triangle sets. In this paper, an improved linear programming lower bound on \(M(I,J)\) is established. In addition, new optimal constructions are presented when \((I,J)\) is one of \((2, 9)\), \((3, 7)\), \((5, 5)\), and \((9, 4)\).