A \((w;\rho_ 2;k,\lambda)\) balanced ternary design (BTD) on a \(w\)-set \(W\) with a hole of size \(v\) consists of a \(v\)-subset \(V\) of \(W\) (called the hole) together with a collection of \(k\)-submultisets of \(W\) (called blocks) such that each element in \(W\backslash V\) appears 0, 1, or 2 times in each block (precisely 2 times in \(\rho_ 2\) blocks), at most one element (counting repetitions) of each block is in \(V\), and each 2- subset of \(W\) not contained in \(V\) appears \(\lambda\) times in the collection. The author establishes the existence of a \((2v-\rho_ 2+1;\rho_ 2;3,2) \text{BTD}\) and a \((2v-\rho_ 2+4;\rho_ 2;3,2) \text{BTD}\), each with a hole of size \(v\), for all positive integers \(v\) and \(\rho_ 2\) where \(v>2\rho_ 2\). This result is then used to establish the number of blocks that two simple \((w;\rho_ 2;3,2)\) BTD's on a common \(w\)-set (and with empty holes) can have in common, for \(\rho_ 2=3\) or 4. The results for \(\rho_ 2=1\) or 2 have been established recently as well by E. J. Billington et al.