# Completion and deficiency problems

## TL;DR

This paper investigates the minimal size of complete Steiner triple systems containing a given partial system, introduces the concept of deficiency in graphs relative to spanning properties, and explores related problems in combinatorial design theory.

## Contribution

It provides asymptotically optimal bounds for embedding sparse partial Steiner triple systems into complete ones and introduces the new framework of deficiency problems for various combinatorial properties.

## Key findings

- Partial STS of size up to r can be embedded into a complete STS of order n+O(√r).
- Introduces the concept of deficiency for graphs with respect to spanning properties.
- Proposes systematic study and future directions for deficiency problems in combinatorics.

## Abstract

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 $r \le \varepsilon n^2$ 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 of the graph $G$ with respect to the property $\mathcal{P}$ to be the smallest positive integer $t$ such that the join $G\ast K_t$ 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.