Yes, it is always possible to achieve these exact bounds. This was done in a slightly different language in

Th. Skolem, "On Certain Distributions of Integers in Pairs with Given Differences", Mathematica Scandinavica 5 (1957), 57–68

For example, when $L=4k$ you can use the decomposition

• $(4k-2i,8k+i, 12k-i)$ for $0\le i\le 2k-1$
• $(4k-2i-1, 4k+i,8k-1-i)$ for $1\le i\le k-1$
• $(2k-2i-1, 5k+2+i,7k-1-i)$ for $0\le i\le k-3$
• $(1, 5k,5k+1)$ and $(4k-1, 6k+1, 10k)$

In order to translate Skolem's partition like the example I did above, write your triples in the form $(i,L+a_i, L+b_i)$ for $1\le i \le L$ and use the pairs $(a_i,b_i)$ from the paper above.

Yes, it is always possible to achieve these exact bounds. This was done in a slightly different language in

Th. Skolem, "On Certain Distributions of Integers in Pairs with Given Differences", Mathematica Scandinavica 5 (1957), 57–68

For example, when $L=4k$ you can use the decomposition

• $(4k-2i,8k+i, 12k-i)$ for $0\le i\le 2k-1$
• $(4k-2i-1, 4k+i,8k-1-i)$ for $1\le i\le k-1$
• $(2k-2i-3, 5k+2+i,7k-1-i)$ for $0\le i\le k-3$
• $(1, 5k,5k+1), (2k-1, 6k, 8k-1)$ and $(4k-1, 6k+1, 10k)$

In order to translate Skolem's partition like the example I did above, write your triples in the form $(i,L+a_i, L+b_i)$ for $1\le i \le L$ and use the pairs $(a_i,b_i)$ from the paper above.

Yes, it is always possible to achieve these exact bounds. This was done in a slightly different language in

Th. Skolem, "On Certain Distributions of Integers in Pairs with Given Differences", Mathematica Scandinavica 5 (1957), 57–68

For example, when $L=4k$ you can use the decomposition

• $(4k-2i,8k+i, 12k-i)$ for $0\le i\le 2k-1$
• $(4k-2i-1, 4k+i,8k-1-i)$ for $1\le i\le k-1$
• $(2k-2i-1, 5k+2+i,7k-1-i)$ for $0\le i\le k-3$
• $(1, 5k,5k+1)$ and $(4k-1, 6k+1, 10k)$

Yes, it is always possible to achieve these exact bounds. This was done in a slightly different language in

Th. Skolem, "On Certain Distributions of Integers in Pairs with Given Differences", Mathematica Scandinavica 5 (1957), 57–68

For example, when $L=4k$ you can use the decomposition

• $(4k-2i,8k+i, 12k-i)$ for $0\le i\le 2k-1$
• $(4k-2i-1, 4k+i,8k-1-i)$ for $1\le i\le k-1$
• $(2k-2i-1, 5k+2+i,7k-1-i)$ for $0\le i\le k-3$
• $(1, 5k,5k+1)$ and $(4k-1, 6k+1, 10k)$

In order to translate Skolem's partition like the example I did above, write your triples in the form $(i,L+a_i, L+b_i)$ for $1\le i \le L$ and use the pairs $(a_i,b_i)$ from the paper above.