On partial parallel classes in partial Steiner triple systems

Abstract

For an integer $\rho$ such that $1\le \rho \le v∕3$, define $\beta (\rho ,v)$ to be the maximum number of blocks in any partial Steiner triple system on $v$ points in which the maximum partial parallel class has size $\rho$. We obtain lower bounds on $\beta (\rho ,v)$ by giving explicit constructions, and upper bounds on $\beta (\rho ,v)$ result from counting arguments. We show that $\beta (\rho ,v)\in \Theta \left(v\right)$ if $\rho$ is a constant, and $\beta (\rho ,v)\in \Theta \left(v^2\right)$ if $\rho =v∕c$, where $c$ is a constant. When $\rho$ is a constant, our upper and lower bounds on $\beta (\rho ,v)$ differ by a constant that depends on $\rho$. Finally, we apply our results on $\beta (\rho ,v)$ to obtain infinite classes of sequenceable partial Steiner triple systems.

Introduction

Suppose $v$ is a positive integer. A partial Steiner triple system of order $v$, denoted PSTS$\left(v\right)$, is a pair $\mathcal{S}=(X,\mathcal{B})$, where $X$ is a set of $v$ points and $\mathcal{B}$ is a set of 3-subsets of $X$, called blocks, such that every pair of points occurs in at most one block. The number of blocks in $\mathcal{B}$ is usually denoted by $b$. If every pair of points occurs in exactly one block, the PSTS$\left(v\right)$ is a Steiner triple system of order $v$, denoted STS$\left(v\right)$. It is clear that a PSTS$\left(v\right)$ is an STS$\left(v\right)$ if and only if $b=v(v-1)∕6$. An STS$\left(v\right)$ exists if and only if $v\equiv 1,3mod6$. Colbourn and Rosa [8] is the definitive reference for Steiner triple systems and related structures.

Let $D\left(v\right)$ denote the maximum number of blocks in a PSTS$\left(v\right)$. The following well-known theorem gives the values of $D\left(v\right)$ for all positive integers $v$. (For a discussion of this result, see [8].)

Theorem 1.1

Suppose $v\ge 3$. Then

$$D\left(v\right)=\left\{\begin{array}{cc}⌊\frac{v}{3}⌊\frac{v-1}{2}⌋⌋\hfill & \text{if }v⁄\equiv 5mod6\hfill \\ ⌊\frac{v}{3}⌊\frac{v-1}{2}⌋⌋-1\hfill & \text{if }v\equiv 5mod6\text{.}\hfill \end{array}\right.$$

Suppose $\mathcal{S}=(X,\mathcal{B})$ is a PSTS$\left(v\right)$. For any point $x\in X$, the degree of $x$, denoted $d_x$, is the number of blocks in $\mathcal{B}$ that contain $x$. A PSTS$\left(v\right)$ is $d$-regular if $d_x=d$ for all $x\in X$. It is clear that $d_x\le (v-1)∕2$ for all $x$. Also, an STS$\left(v\right)$ is a $d$-regular PSTS$\left(v\right)$ with $d=(v-1)∕2$.

Suppose $\mathcal{S}=(X,\mathcal{B})$ is a PSTS$\left(v\right)$. A parallel class in $\mathcal{S}$ is a set of $v∕3$ disjoint blocks in $\mathcal{B}$. A partial parallel class (or PPC) in $\mathcal{S}$ is any set of disjoint blocks in $\mathcal{B}$. The size of a PPC is the number of blocks in the PPC. A PPC in $\mathcal{S}$ of size $\rho$ is maximum if there does not exist any PPC in $\mathcal{S}$ of size $\rho +1$. Note that a maximum partial parallel class in $\mathcal{S}$ is not necessarily unique, but the size of a maximum PPC is well-defined.

There has been considerable study concerning the sizes of maximum partial parallel classes in STS$\left(v\right)$. It seems to be quite difficult to find STS$\left(v\right)$ with $v\equiv 3mod6$ that do not have parallel classes. It was only in 2015 that the first infinite classes were found, by Bryant and Horsley [7]. For STS$\left(v\right)$ with $v\equiv 1mod6$ that do not have partial parallel classes of size $(v-1)∕3$, there are two infinite classes known, one of which was found by Wilson (see [13]) and one discovered by Bryant and Horsley [6].

There are also lower bounds on the sizes of maximum partial parallel classes in STS$\left(v\right)$. The first nontrivial bound is due to Lindner and Phelps [11], who proved in 1978 that every STS$\left(v\right)$ with $v>27$ has a partial parallel class of size at least $\frac{v-1}{4}$. Improvements have been found by Woolbright [14] and Brouwer [5]. The result proven in [5] is that every STS$\left(v\right)$ has a PPC of size at least $\frac{v}{3}-\frac{5}{3}{v}^{2∕3}$. In 1997, Alon, Kim and Spencer [1] used probabilistic methods to prove that any STS$\left(v\right)$ contains a PPC of size at least $\frac{v}{3}-O\left(v^{1∕2}{(lnv)}^{3∕2}\right)$. Also, a recent preprint by Keevash, Pokrovskiy, Sudakov and Yepremyan [9] improves this lower bound to $\frac{v}{3}-O\left(\frac{logv}{loglogv}\right)$.

We should also mention Brouwer’s conjecture, made in [5], that any STS$\left(v\right)$ has a PPC of size at least $(v-4)∕3$.

The problem of determining lower bounds on the sizes of maximum partial parallel classes in partial Steiner triple systems has received less study. It was shown in [1] that a $d$-regular PSTS$\left(v\right)$ has a PPC of size at least $\frac{v}{3}-O\left(\frac{v{(lnd)}^{3∕2}}{d^{1∕2}}\right)$.

In this paper, we address the following problem. For an integer $\rho$ such that $1\le \rho \le v∕3$, define $\beta (\rho ,v)$ to be the maximum number of blocks in any PSTS$\left(v\right)$ in which the maximum partial parallel class has size $\rho$. We obtain lower bounds on $\beta (\rho ,v)$ by giving explicit constructions, and upper bounds result from counting arguments.

Before continuing, we observe that, when $v\equiv 1,3mod6$, the values $\beta (\lfloor \frac{v}{3}\rfloor ,v)$ follow easily from known results concerning STS$\left(v\right)$.

Theorem 1.2

For all $v\equiv 1,3mod6$, $v\ne 7$, it holds that $\beta (\lfloor \frac{v}{3}\rfloor ,v)=v(v-1)∕6$. Also, $\beta (2,7)=5$.

Proof

For all $v\equiv 3mod6$, it is known that there exists an STS$\left(v\right)$ containing a parallel class. (For example, we can use a Kirkman triple system of order $v$; see [8].) For all $v\equiv 1mod6$, $v\ne 7$, there exists an STS$\left(v\right)$ containing a PPC of size $(v-1)∕3$ (For $v\ge 19$, we can use a nearly Kirkman triple system of order $v$; see [8]. For $v=13$, it is noted in [8] that both of the two nonisomorphic STS$\left(13\right)$ have a PPC of size $4$.) Finally, for $v=7$, the PSTS$\left(7\right)$ consisting of blocks $123,456,147,257,367$ has a maximum PPC of size $2$, and there is no PSTS$\left(7\right)$ having six blocks that has a maximum PPC of size $2$.  □

As mentioned above, there are STS$\left(v\right)$ in which the maximum partial parallel class has size less than $\lfloor v∕3\rfloor -1$. In this situation, we would have $\beta (\rho ,v)=v(v-1)∕6$, where $\rho$ is the size of the maximum PPC in the given STS$\left(v\right)$. For example, there is an STS$\left(15\right)$ having a maximum PPC of size $4$, as well as an STS$\left(21\right)$ having a maximum PPC of size $6$. Therefore, $\beta (4,15)=35$ and $\beta (6,21)=70$.

We summarize the main contributions of this paper. In Section 2, we present constructions for partial Steiner triple systems that have maximum PPCs of a prespecified size, thus obtaining lower bounds on $\beta (\rho ,v)$. In Section 3, we prove an upper bound on $\beta (\rho ,v)$ using a counting argument. In Section 4, we show that $\beta (\rho ,v)\in \Theta \left(v\right)$ if $\rho$ is a constant, and $\beta (\rho ,v)\in \Theta \left(v^2\right)$ if $\rho =v∕c$, where $c$ is a constant. When $\rho$ is a constant, our upper and lower bounds on $\beta (\rho ,v)$ differ by a constant that depends on $\rho$. In Section 5, we apply our results to the problem of finding sequenceable PSTS$\left(v\right)$. In particular, the constructions we describe in Section 2 provide infinite classes of sequenceable PSTS$\left(v\right)$. Finally, Section 6 is a brief summary.