On partial parallel classes in partial Steiner triple systems

For an integer ρ such that 1≤ρ≤v∕3, define β(ρ,v) to be the maximum number of blocks in any partial Steiner triple system on v points which parallel class has size ρ. We obtain lower bounds by giving explicit constructions, and upper result from counting arguments. show β(ρ,v)∈Θ(v) if is a constant, β(ρ,v)∈Θ(v2) ρ=v∕c, where c constant. When our differ constant depends Finally, we apply results infinite classes sequenceable systems.