A Note on Embedding Partial Extended Triple Systems of Even Index

Recently, Raines and Rodger have proved that for all \(\lambda \geq 1\), any partial extended triple system of order \(n\) and index \(\lambda\) can be embedded in a (complete) extended triple system of order \(v\) and index \(\lambda\) for any even \(v \geq 4n + 6\). In this note, it is shown that if \(\lambda\) is even then this bound on \(v\) can be improved to all \(v \geq 3n + 5\), and under some conditions to all \(v \geq 2n + 1\).