Given a partial Steiner triple system (STS) of order n, what is the order of the smallest complete STS it can be embedded into? The study of this question goes back more than 40 years. In this paper we answer it for relatively sparse STSs, showing that given a partial STS of order n with at most $r\le \epsilon n^2$ triples, it can always be embedded into a complete STS of order $n+O(\sqrt{r})$, which is asymptotically optimal. We also obtain similar results for completions of Latin squares and other designs.

This suggests a new, natural class of questions, called deficiency problems. Given a global spanning property $\mathcal{P}$ and a graph G, we define the deficiency $\text{def}(G)$ of the graph G with respect to the property $\mathcal{P}$ to be the smallest positive integer t such that the join $G⁎K_t$ has property $\mathcal{P}$. To illustrate this concept we consider deficiency versions of some well-studied properties, such as having a $K_k$-decomposition, Hamiltonicity, having a triangle-factor and having a perfect matching in hypergraphs.

The main goal of this paper is to propose a systematic study of these problems; thus several future research directions are also given.

给定阶数为n的部分Steiner三元系统（STS），可以嵌入到其中的最小完整STS的阶数是多少？这个问题的研究可以追溯到40年前。在本文中，我们针对相对稀疏的STS回答了这一问题，表明给定了n阶的部分STS，最多$\mathrm{[R}\le \epsilon {ñ}^2$ 三元组，它始终可以嵌入到完整的订单STS中 $ñ+Ø（\sqrt{\mathrm{[R}}）$，这是渐近最优的。对于拉丁方和其他设计的完成，我们也获得类似的结果。

这提出了一种新的自然问题，称为缺陷问题。拥有全球性的财产$\mathcal{P}$和图G，我们定义了缺陷$\text{定义}（G）$该图的G ^相对于该属性$\mathcal{P}$是最小的正整数t，使得连接$G⁎{ķ}_{Ť}$ 有财产 $\mathcal{P}$。为了说明这个概念，我们考虑一些经过充分研究的属性的不足版本，例如具有${ķ}_{ķ}$分解，具有汉密尔顿性，具有三角因子并且在超图中具有完美匹配。

本文的主要目的是对这些问题进行系统的研究。因此也给出了一些未来的研究方向。