Landang Yuan

An Steiner triple system of order $v$, denoted by $STS(v)$, is a pair $(X, {\cal B})$, where $X$ is a $v$-set, ${\cal B}$ is a collection of triples on $X$, such that each 2-set of $X$ occurs in exactly one triple of ${\cal B}$. An $STS(v)=(X,{\cal B})$ is resolvable, denoted by $KTS(v)$, if there exists a partition $\Gamma=\{P_1,P_2,\cdot\cdot\cdot,P_{\frac{v-1}{2}}\}$ of ${\cal B}$ such that each part $P_i$, called parallel class, forms a partition of $X$. It is well known that there exists a $KTS(v)$ if and only if $v\equiv3$ mod 6.
An overlarge set of Kirkman triple systems, denoted by $OLKTS(v)$, is a collection $\{(X\backslash\{x\},{\cal B}_x)\colon x\in X\}$, where $X$ is a $(v+1)$-set, each $(X\backslash\{x\},{\cal B}_x)$ is a $KTS(v)$, and these ${\cal B}_x,~x\in X$, partition all triples on $X$. The existence of $OLKTS(v)$ has been investigated. The known results are that there exists an $OLKTS(v)$ for $v=9,~4^n-1$ and $2q^n+1$, where $n \geq 1$ and prime power $q=13$ or $q\equiv 7~({\rm mod ~12})$.
In this paper, in order to study $OLKTS(v)$, we introduce a kind of auxiliary designs---overlarge sets of generalized Kirkman systems, and obtain some results for its existence. Our main conclusion is to exist an $OLKTS(6u+3)$ for $u=2^{2n+1}-1,~7^n,~31^n$ and $127^n$, where $n\geq1$.