On transversal and 2-packing numbers in uniform linear systems

TL;DR

This paper investigates the relationship between transversal and 2-packing numbers in uniform linear systems, providing new constructions, properties, and characterizations, especially for intersecting systems where these parameters are equal.

Contribution

It introduces an infinite family of linear systems with equal transversal and 2-packing numbers, and characterizes 4-uniform intersecting systems with this property.

Findings

Constructed an infinite family with $ au= u_2$ and minimal size

Analyzed properties of $r$-uniform intersecting systems with $ au= u_2=r$

Characterized 4-uniform intersecting systems with $ au= u_2=4$

Abstract

A linear system is a pair $(P,\mathcal{L})$ where $\mathcal{L}$ is a family of subsets on a ground finite set $P$, such that $|l\cap l^\prime|\leq 1$, for every $l,l^\prime \in \mathcal{L}$. The elements of $P$ and $\mathcal{L}$ are called points and lines, respectively, and the linear system is called intersecting if any pair of lines intersect in exactly one point. A subset $T$ of points of $P$ is a transversal of $(P,\mathcal{L})$ if $T$ intersects any line, and the transversal number, $\tau(P,\mathcal{L})$, is the minimum order of a transversal. On the other hand, a 2-packing set of a linear system $(P,\mathcal{L})$ is a set $R$ of lines, such that any three of them have a common point, then the 2-packing number of $(P,\mathcal{L})$, $\nu_2(P,\mathcal{L})$, is the size of a maximum 2-packing set. It is known that the transversal number $\tau(P,\mathcal{L})$ is bounded above by a quadratic function of $\nu_2(P,\mathcal{L})$. An open problem is to haracterize the families of linear systems which satisfies $\tau(P,\mathcal{L})\leq \lambda\nu_2(P,\mathcal{L})$, for some $\lambda\geq1$. In this paper, we give an infinite family of linear systems $(P,\mathcal{L})$ which satisfies $\tau(P,\mathcal{L})=\nu_2(P,\mathcal{L})$ with smallest possible cardinality of $\mathcal{L}$, as well as some properties of $r$-uniform intersecting linear systems $(P,\mathcal{L})$, such that $\tau(P,\mathcal{L})=\nu_2(P,\mathcal{L})=r$. Moreover, we state a characterization of $4$-uniform intersecting linear systems $(P,\mathcal{L})$ with $\tau(P,\mathcal{L})=\nu_2(P,\mathcal{L})=4$.

Figures

Equations

$$\lceil {\nu }_2/2\rceil \le \tau \le \frac{{\nu }_2({\nu }_2-1)}{2}.$$

$$\{l_p\cap L_i:i=1,\dots ,\Delta -1\}\cup \{a_{\Delta },\dots ,a_{{\nu }_2-2}\}\cup \{p\},$$

$$\mathcal{L}=\{L_g:g\in \Gamma \setminus \{e\}\},\text{mbox}where{L}_g=\{(h,g):h\in \Gamma \},$$

$${\mathcal{L}}_p=\{l_{p_g}:g\in \Gamma \},\text{mbox}where{l}_{p_g}=\{(g,h):h\in \Gamma \setminus \{e\}\}\cup \{p\},$$

$$l_{q_{g}}=\{(h,f_{g}(h)):h\in\Gamma,f_{g}(h)=h+g\mbox{ with $f_{g}(h)\neq e$}\}\cup\{q\},$$

$$\mathcal{L}$$

$${\mathcal{L}}_p$$

$${\mathcal{L}}_q$$

$$\tau ({\mathcal{C}}_{n,n+1})=n+1$$

$$T=\{(h_0,f_{g_0}(h_0)),\dots ,(h_{n-1},f_{g_{n-1}}(h_{n-1}))\},$$

$$\sum _{i=0}^{n-1}{f}_{h_i}(g_i)=\sum _{i=0}^{n-1}(g_i+h_i)=\sum _{i=0}^{n-1}{g}_i+\sum _{i=0}^{n-1}{h}_i=e,$$

$${\nu }_2({\mathcal{C}}_{n,n+1})=n+1$$

$$l_{p_i}$$

$$l_{q_j}$$

$$L_k$$

$$\tau ({\mathcal{C}}_{n,n+1})={\nu }_2({\mathcal{C}}_{n,n+1})=n+1,$$

$$3n^2-n=|E({\mathcal{C}}_{n,n+1})|>\frac{3(n^2+2n-1)}{2},$$

$$3(r-1)\le |\mathcal{L}|\le r^2-r+1.$$

Full text

Keywords. Linear systems, transversal number, 2-packing number, finite projective plane.

1 Introduction

A linear system is a pair $(P,\mathcal{L})$ where $\mathcal{L}$ is a family of subsets on a ground finite set $P$, such that $|l\cap l^{\prime}|\leq 1$, for every pair of distinct subsets $l,l^{\prime}\in\mathcal{L}$. The linear system $(P,\mathcal{L})$ is intersecting if $|l\cap l^{\prime}|=1$, for every pair of distinct subsets $l,l^{\prime}\in\mathcal{L}$. The elements of $P$ and $\mathcal{L}$ are called points and lines, respectively; a line with exactly $r$ points is called a $r$-line, and the rank of $(P,\mathcal{L})$ is the maximum cardinality of a line in $(P,\mathcal{L})$, when all the lines of $(P,\mathcal{L})$ are $r$ lines we have a $r$-uniform linear system. In this context, a simple graph is an 2-uniform linear system.

A subset $T\subseteq P$ is a transversal (also called vertex cover or hitting set in many papers, as example [7, 9, 11, 12, 14, 16, 17, 18, 19, 20, 21]) of $(P,\mathcal{L})$ if for any line $l\in\mathcal{L}$ satisfies $T\cap l\neq\emptyset$. The transversal number of $(P,\mathcal{L}),$ denoted by $\tau(P,\mathcal{L}),$ is the smallest possible cardinality of a transversal of $(P,\mathcal{L})$.

A subset $R\subseteq\mathcal{L}$ is called 2-packing of $(P,\mathcal{L})$ if three elements are chosen in $R$ then they are not incident in a common point. The 2-packing number of $(P,\mathcal{L})$, denoted by $\nu_{2}(P,\mathcal{L})$, is the maximum number of a 2-packing of $(P,\mathcal{L})$.

There are many interesting works studying the relationship between these two parameters, for instance, in [20], the authors propose the problem of bounding $\tau(P,\mathcal{L})$ in terms of a function of $\nu_{2}(P,\mathcal{L})$ for any linear system. In [2], some authors of this paper and others proved that any linear system satisfies:

[TABLE]

That is, the transversal number, $\tau$, of any linear system is upper bounded by a quadratic function of their 2-packing number, $\nu_{2}$.

In order to find how a function of $\nu_{2}(P,\mathcal{L})$ can bound $\tau(P,\mathcal{L})$, the authors of [10] using probabilistic methods to prove that $\tau\leq\lambda\nu_{2}$ does not hold for any positive $\lambda$. In particular, they exhibit the existence of $k$-uniform linear systems $(P,\mathcal{L})$ for which their transversal number is $\tau(P,\mathcal{L})=n-o(n)$ and their $2$-packing number is upper bounded by $\frac{2n}{k}$.

Nevertheless, there are some relevant works about families of linear systems in which their transversal numbers are upper bounded by a linear function of their 2-packing numbers. In [1] the authors proved that if $(P,\mathcal{L})$ is a 2-uniform linear system, a simple graph, with $|\mathcal{L}|>\nu_{2}(P,\mathcal{L})$ then $\tau(P,\mathcal{L})\leq\nu_{2}(P,\mathcal{L})-1$; moreover, they characterize the simple connected graphs that attain this upper bound and the lower bound given in Equation (1). In [2] was proved that the linear systems $(P,\mathcal{L})$ with $|\mathcal{L}|>\nu_{2}(P,\mathcal{L})$ and $\nu_{2}(P,\mathcal{L})\in\{2,3,4\}$ satisfy $\tau(P,\mathcal{L})\leq\nu_{2}(P,\mathcal{L})$; and when attain the equality, they are a special family of linear subsystems of the projective plane of order $3$, $\Pi_{3}$, with transversal and $2$-packing numbers equal to $4$. Moreover, they proved that $\tau(\Pi_{q})\leq\nu_{2}(\Pi_{q})$ when $\Pi_{q}=(P_{q},\mathcal{L}_{q})$ is a projective plane of order $q$, consequently the equality holds when q is odd.

The rest of this paper is structured as follows: In Section 2, we present a result about linear systems satisfying $\tau\leq\nu_{2}-1$. In Section 3, we give an infinite family of linear systems such that $\tau=\nu_{2}$ with smallest possible cardinality of lines. And, finally, in the last section, we presented some properties of the $r$-uniform linear systems, such that $\tau=\nu_{2}=r$, and we characterize the $4$-uniform linear systems with $\tau=\nu_{2}=4$.

2 On linear systems with $\tau\leq\nu_{2}-1$

Let $(P,\mathcal{L})$ be a linear system and $p\in P$ be a point. It is denoted by $\mathcal{L}_{p}$ to the set of lines incident to $p$. The degree of $p$ is defined as $deg(p)=|\mathcal{L}_{p}|$ and the maximum degree overall points of the linear systems is denoted by $\Delta(P,\mathcal{L})$. A point of degrees $2$ and $3$ is called double and triple point, respectively, and two points $p$ and $q$ in $(P,\mathcal{L})$ are adjacent if there is a line $l\in\mathcal{L}$ with $\{p,q\}\subseteq l$.

In this section, we generalize Proposition 2.1, Proposition 2.2, Lemma 2.1, Lemma 3.1 and Lemma 4.1 of [2] proving that a linear system $(P,\mathcal{L})$ with $|\mathcal{L}|>\nu_{2}(P,\mathcal{L})$ and “few” lines satisfies $\tau(P,\mathcal{L})\leq\nu_{2}(P,\mathcal{L})-1$. Notice that, through this paper, all linear systems $(P,\mathcal{L})$ are considered with $|\mathcal{L}|>\nu_{2}(P,\mathcal{L})$ due to the fact $|\mathcal{L}|=\nu_{2}(P,\mathcal{L})$ if and only if $\Delta(P,\mathcal{L})\leq 2$.

Theorem 2.1**.**

Let $(P,\mathcal{L})$ be a linear system with $p,q\in P$ be two points such that $deg(p)=\Delta(P,\mathcal{L})$ and $deg(q)=\max\{deg(x):x\in P\setminus\{p\}\}$. If $|\mathcal{L}|\leq deg(p)+deg(q)+\nu_{2}(P,\mathcal{L})-3$, then $\tau(P,\mathcal{L})\leq\nu_{2}(P,\mathcal{L})-1$.

Proof

Let $p,q\in P$ be two points as in the theorem, and let $\mathcal{L}^{\prime\prime}=\mathcal{L}\setminus\{\mathcal{L}_{p}\cup\mathcal{L}_{q}\}$, which implies that $|\mathcal{L}^{\prime\prime}|\leq\nu_{2}(P,\mathcal{L})-2$. Assume that $|\mathcal{L}^{\prime\prime}|=\nu_{2}(P,\mathcal{L})-2$ ($\mathcal{L}_{p}\cap\mathcal{L}_{q}\neq\emptyset$), otherwise, the following set $\{p,q\}\cup\{a_{l}:a_{l}\mbox{ is any point of }l\in\mathcal{L}^{\prime\prime}\}$ is a transversal of $(P,\mathcal{L})$ of cardinality at most $\nu_{2}(P,\mathcal{L})-1$, and the statement holds. Suppose that $\mathcal{L}^{\prime\prime}=\{L_{1},\ldots,L_{\nu_{2}-2}\}$ is a set of pairwise disjoint lines because, in otherwise, they induce at least a double point, $x\in P$, hence the following set of points $\{p,q,x\}\cup\{a_{l}:l\in\mathcal{L}^{\prime\prime}\setminus\mathcal{L}^{\prime\prime}_{x}\}$, where $a_{l}$ is any point of $l$, is a transversal of $(P,\mathcal{L})$ of cardinality at most $\nu_{2}(P,\mathcal{L})-1$, and the statement holds. Let $l_{q}\in\mathcal{L}_{q}\setminus\{l_{p,q}\}$ be a fixed line and let $l_{p}$ be any line of $\mathcal{L}_{p}\setminus\{l_{p,q}\}$, where $l_{p,q}$ is the line containing to $p$ and $q$ (since $\mathcal{L}_{p}\cap\mathcal{L}_{q}\neq\emptyset$). Then $l_{p}\cap l_{q}\neq\emptyset$, since the $l_{q}$ induce a triple point on the following 2-packing $\mathcal{L}^{\prime\prime}\cup\{l_{p},l_{p,q}\}$, which implies that there exists a line $L_{p,q}\in\mathcal{L}^{\prime\prime}$ with $l_{q}\cap l_{p}\cap L_{p,q}\neq\emptyset$, and hence $l_{p}\cap l_{q}\neq\emptyset$. Consequently, $deg(q)=\Delta(P,\mathcal{L})$ and $\Delta(P,\mathcal{L})\leq\nu_{2}(P,\mathcal{L})-1$ (since $\deg(p)-1\leq\nu_{2}(P,\mathcal{L})-2$). Therefore, the following set:

[TABLE]

where $a_{i}$ is any point of $L_{i}$, for $i=\Delta,\ldots,\nu_{2}-2$, is a transversal of $(P,\mathcal{L})$ of the cardinality at most $\nu_{2}(P,\mathcal{L})-1$, and the statement holds.∎

3 A family of uniform linear systems with $\tau=\nu_{2}$

In this section, we exhibit an infinite family of linear systems $(P,\mathcal{L})$ with two points of maximum degree and $|\mathcal{L}|=2\Delta(P,\mathcal{L})+\nu_{2}(P,\mathcal{L})-2$ with $\tau(P,\mathcal{L})=\nu_{2}(P,\mathcal{L})$. It is immediately, by Theorem 2.1, that $\tau(P,\mathcal{L})\leq\nu_{2}(P,\mathcal{L})-1$ for linear systems with less lines.

In the remainder of this paper, $(\Gamma,+)$ is an additive Abelian group with neutral element $e$. Moreover, if $\sum_{g\in\Gamma}g=e$, then the group is called neutral sum group. In the following, every group $(\Gamma,+)$ is a neutral sum group, such that $2g\neq e$, for all $g\in\Gamma\setminus\{e\}$. As an example of this type of groups we have $(\mathbb{Z}_{n},+)$, for $n\geq 3$ odd.

Let $n=2k+1$, with $k$ a positive integer, and $(\Gamma,+)$ be a neutral sum group of order $n$. Let:

[TABLE]

for $g\in\Gamma\setminus\{e\}$, and:

[TABLE]

for $g\in\Gamma$, and $\mathcal{L}_{q}=\{l_{q_{g}}:g\in\Gamma\}$, where:

[TABLE]

for $g\in\Gamma$.

Hence, the set of lines $\mathcal{L}$ is a set of pairwise disjoint lines with $|\mathcal{L}|=n-1$ and each line of $\mathcal{L}$ has $n$ points. On the other hand, $\mathcal{L}_{p}$ and $\mathcal{L}_{q}$ are set of lines incidents to $p$ and $q$, respectively, with $|\mathcal{L}_{p}|=|\mathcal{L}_{p}|=n$, and each line of $\mathcal{L}_{p}\cup\mathcal{L}_{q}$ has $n$ points. Moreover, this set of lines satisfies that, giving $l_{p_{a}}\in\mathcal{L}_{p}$ there exists an unique $l_{q_{b}}\in\mathcal{L}_{q}$ with $l_{p_{a}}\cap l_{q_{b}}=\emptyset$, otherwise, there exits $l_{p_{a}}\in\mathcal{L}_{p}$ such that $l_{p_{a}}\cap l_{q_{b}}\neq\emptyset$, for all $l_{q_{b}}\in\mathcal{L}_{q}$, which implies that $a+b\in\Gamma\setminus\{e\}$, for all $b\in\Gamma$, which is a contradiction.

The linear system $(P_{n},\mathcal{L}_{n})$ with $P_{n}=(\Gamma\times\Gamma\setminus\{e\})\cup\{p,q\}\quad\!\!\!\!\mbox{and}\quad\!\!\!\!\mathcal{L}_{n}=\mathcal{L}\cup\mathcal{L}_{p}\cup\mathcal{L}_{q}$, denoted by $\mathcal{C}_{n,n+1}$, is an $n$-uniform linear system with $n(n-1)+2$ points and $3n-1$ lines. Notice that, this linear system has 2 points of degree $n$ (points $p$ and $q$) and $n(n-1)$ points of degree $3$.

A linear subsystem $(P^{\prime},\mathcal{L}^{\prime})$ of a linear system $(P,\mathcal{L})$ satisfies that for any line $l^{\prime}\in\mathcal{L}^{\prime}$ there exists a line $l\in\mathcal{L}$ such that $l^{\prime}=l\cap P^{\prime}$, where $P^{\prime}\subset P$. Given a linear system $(P,\mathcal{L})$ and a point $p\in P$, the linear system obtained from $(P,\mathcal{L})$ by deleting the point $p$ is the linear system $(P^{\prime},\mathcal{L}^{\prime})$ induced by $\mathcal{L}^{\prime}=\{l\setminus\{p\}:l\in\mathcal{L}\}$. On the other hand, given a linear system $(P,\mathcal{L})$ and a line $l\in\mathcal{L}$, the linear system obtained from $(P,\mathcal{L})$ by deleting the line $l$ is the linear system $(P^{\prime},\mathcal{L}^{\prime})$ induced by $\mathcal{L}^{\prime}=\mathcal{L}\setminus\{l\}$. The linear systems $(P,\mathcal{L})$ and $(Q,\mathcal{M})$ are isomorphic, denoted by $(P,\mathcal{L})\simeq(Q,\mathcal{M})$, if after deleting the points of degree 1 or 0 from both, the systems $(P,\mathcal{L})$ and $(Q,\mathcal{M})$ are isomorphic as hypergraphs (see [4]).

It is important to state that in the rest of this paper it is considered linear systems $(P,\mathcal{L})$ without points of degree one because, if $(P,\mathcal{L})$ is a linear system which has all lines with at least two points of degree 2 or more, and $(P^{\prime},\mathcal{L}^{\prime})$ is the linear system obtained from $(P,\mathcal{L})$ by deleting all points of degree one, then they are essentially the same linear system because it is not difficult to prove that transversal and 2-packing numbers of both coincide (see [2]).

Example 3.1**.**

Let $\Gamma=\mathbb{Z}_{3}$. The linear system $\mathcal{C}_{3,4}=(P_{3},\mathcal{L}_{3})$ has as set of points to $P_{3}=\{(0,1),(1,1),(2,1),(0,2),(1,2),(2,2)\}\cup\{p\}\cup\{q\}$ and as set of lines to $\mathcal{L}_{3}=\mathcal{L}\cup\mathcal{L}_{p}\cup\mathcal{L}_{q}$, where

[TABLE]

and depicted in Figure 1. This linear system is isomorphic to the linear system giving in [2] Figure 3, which is the linear system with the less number of lines and maximum degree 3 such that $\tau=\nu_{2}=4$.

Proposition 3.1**.**

The linear system $\mathcal{C}_{n,n+1}$ satisfies that:

[TABLE]

Proof

Notice that $\tau(\mathcal{C}_{n,n+1})\leq n+1$ since \{x_{g}:\mbox{ x_{g}$is any point of$L_{g}\in\mathcal{L}}\}\cup\{p,q\} is a transversal of $\mathcal{C}_{n,n+1}$. To prove that $\tau(P_{n},\mathcal{L}_{n})\geq n+1$, suppose on the contrary that $\tau(P_{n},\mathcal{L}_{n})=n$. If $T$ is a transversal of cardinality $n$ then $T\subseteq\Gamma\times\Gamma\setminus\{e\}$, i.e., $p,q\not\in T$ because, in other case, if $p\in T$ then, by the Pigeonhole principle, there is a line $l_{q_{a}}\in\mathcal{L}_{q}$ such that $T\cap l_{q_{a}}=\emptyset$, since $deg(q)=n$, which is a contradiction, unless that $q\in T$, which implies that there exists $L\in\mathcal{L}$ such that $L\cap T=\emptyset$ (because $|\mathcal{L}|=n-1$), which is also a contradiction. Therefore $T\subseteq\Gamma\times\Gamma\setminus\{e\}$.

Suppose that:

[TABLE]

where $\{h_{0},\ldots,h_{n-1}\}=\{g_{0},\ldots,g_{n-1}\}=\Gamma$ and $f_{g_{i}}=h_{i}+g_{i}\neq e$, for $i=0,\ldots,n-1$. Then:

[TABLE]

since $\displaystyle\sum_{g\in\Gamma}g=\displaystyle\sum_{g\in\Gamma\setminus\{e\}}g=e$, which implies that there exists $f_{h_{j}}(g_{j})\in T$ that satisfies $f_{h_{j}}(g_{j})=e$, which is a contradiction, and consequently $\tau(\mathcal{C}_{n,n+1})=n+1$.∎

Proposition 3.2**.**

The linear system $\mathcal{C}_{n,n+1}$ satisfies that:

[TABLE]

.

Proof

Notice that $\nu_{2}(\mathcal{C}_{n,n+1})\geq n+1$ because, for any two lines $l_{p_{g}},l_{p_{h}}\in\mathcal{L}_{p}$, $\mathcal{L}\cup\{l_{p_{g}},l_{p_{h}}\}$ is a 2-packing. To prove that $\nu_{2}(\mathcal{C}_{n,n+1})\leq n+1$, suppose on the contrary that $\nu_{2}(\mathcal{C}_{n,n+1})=n+2$, and that $R$ is a maximum 2-packing of size $n+2$, we analyze to cases:

Case $(i)$: Suppose that $R=\mathcal{L}\cup\{l_{p_{a}},l_{p_{b}},l_{q_{c}}\}$, where $l_{p_{a}},l_{p_{b}}\in\mathcal{L}_{p}$ and $l_{q_{c}}\in\mathcal{L}_{q}$; since there is an unique line $l_{p}\in\mathcal{L}_{p}$ which intersect to $l_{q_{c}}$, then we assume that $l_{p_{a}}\cap l_{q_{c}}\neq\emptyset$. By construction of $\mathcal{C}_{n,n+1}$ there exits $L\in\mathcal{L}$ that satisfies $l_{p_{a}}\cap l_{q_{c}}\cap L\neq\emptyset$, inducing a triple point, which is a contradiction.

Case $(ii)$: Let $k$ be an element of $\Gamma\setminus\{e\}$ and $R=\{l_{p_{a}},l_{p_{b}},l_{q_{c}},l_{q_{d}}\}\cup\mathcal{L}\setminus\{L_{k}\}$ with $l_{p_{a}},l_{p_{b}}\in\mathcal{L}_{p}$ and $l_{q_{c}},l_{q_{d}}\in\mathcal{L}_{q}$, without loss of generality, suppose that $l_{p_{a}}\cap l_{q_{c}}\neq\emptyset$, $l_{p_{b}}\cap l_{q_{d}}\neq\emptyset$, $l_{p_{a}}\cap l_{q_{d}}=\emptyset$ and $l_{p_{b}}\cap l_{q_{c}}=\emptyset$, otherwise, $R$ is not a 2-packing. It is claimed that there exists $L\in\mathcal{L}\setminus\{L_{k}\}$ such that either $l_{p_{a}}\cap l_{q_{c}}\cap L\neq\emptyset$ or $l_{p_{b}}\cap l_{q_{d}}\cap L\neq\emptyset$, which implies that $R$ induce a triple point, which is contradiction and hence $\nu_{2}(\mathcal{C}_{n,n+1})=n+1$. To verify the claim suppose on the contrary that every $L\in\mathcal{L}\setminus\{L_{k}\}$ satisfies $l_{p_{a}}\cap l_{q_{c}}\cap L=\emptyset$ and $l_{p_{b}}\cap l_{q_{d}}\cap L=\emptyset$. It means that $l_{p_{a}}\cap l_{q_{c}}\cap L_{k}\neq\emptyset$ and $l_{p_{b}}\cap l_{q_{d}}\cap L_{k}\neq\emptyset$. By construction of $\mathcal{C}_{n,n+1}$ it follows that:

[TABLE]

If $l_{p_{a}}\cap l_{q_{c}}\cap L_{k}\neq\emptyset$ and $l_{p_{b}}\cap l_{q_{d}}\cap L_{k}\neq\emptyset$, then $a+c=b+d=k$. On the other hand, as $l_{p_{a}}\cap l_{q_{d}}=\emptyset$ and $l_{p_{b}}\cap l_{q_{c}}=\emptyset$, then $a+d=b+c=e$. As a consequence of $a+c=b+d=k$ and $a+d=b+c=e$ we obtain $2k=e$, which is a contradiction. Therefore, $\nu_{2}(\mathcal{C}_{n,n+1})=n+1$.∎

Hence, by Proposition 3.1 and Proposition 3.2 it was proved that:

Theorem 3.2**.**

Let $n=2k+1$, with $k\in\mathbb{N}$, then

[TABLE]

with smallest possible cardinality of lines.

3.1 Straight line systems

A straight line representation on $\mathbb{R}^{2}$ of a linear system $(P,\mathcal{L})$ maps each point $x\in P$ to a point $p(x)$ of $\mathbb{R}^{2}$, and each line $L\in\mathcal{L}$ to a straight line segment $l(L)$ of $\mathbb{R}^{2}$ in such a way that for each point $x\in P$ and line $L\in\mathcal{L}$ satisfies $p(x)\in l(L)$ if and only if $x\in L$, and for each pair of distinct lines $L,L^{\prime}\in\mathcal{L}$ satisfies $l(L)\cap l(L^{\prime})=\{p(x):x\in L\cap L^{\prime}\}$. A straight line system $(P,\mathcal{L})$ is a linear system, such that it has a straight line representation on $\mathbb{R}^{2}$. In [2] was proved that the linear system $\mathcal{C}_{3,4}$ is not a straight one. The Levi graph of a linear system $(P,\mathcal{L})$, denoted by $B(P,\mathcal{L})$, is a bipartite graph with vertex set $V=P\cup\mathcal{L}$, where two vertices $p\in P$, and $L\in\mathcal{L}$ are adjacent if and only if $p\in L$.

In the same way as in [2] and according to [15], any straight line system is Zykov-planar, see also [23]. Zykov proposed to represent the lines of a set system by a subset of the faces of a planar map on $R^{2}$, i.e., a set system $(X,\mathcal{F})$ is Zykov-planar if there exists a planar graph $G$ (not necessarily a simple graph) such that $V(G)=X$ and $G$ can be drawn in the plane with faces of $G$ two-colored (say red and blue) so that there exists a bijection between the red faces of $G$ and the subsets of $\mathcal{F}$ such that a point $x$ is incident with a red face if and only if it is incident with the corresponding subset. In [22] was shown that the Zykov’s definition is equivalent to the following: A set system $(X,\mathcal{F})$ is Zykov-planar if and only if the Levi graph $B(X,\mathcal{F})$ is planar. It is well-known that for any planar graph $G$ the size of $G$, $|E(G)|$, is upper bounded by $\frac{k(|V(G)|-2)}{k-2}$ (see [5] page 135, exercise 9.3.1 (a)), where $k$ is the girth of $G$ (the length of a shortest cycle contained in the graph $G$). It is not difficult to prove that the Levi graph $B(\mathcal{C}_{n,n+1})$ of $\mathcal{C}_{n,n+1}$ is not a planar graph, since the size of the girth of $B(\mathcal{C}_{n,n+1})$ is $6$, it follows:

[TABLE]

for all $n\geq 3$. Therefore, the linear system $\mathcal{C}_{n,n+1}$ is not a straight line system.

Finally, as a Corollary of Theorem 2.1, we have the following:

Corollary 3.1**.**

Let $(P,\mathcal{L})$ be a straight line system with $p,q\in P$ be two points such that $deg(p)=\Delta(P,\mathcal{L})$ and $deg(q)=\max\{deg(x):x\in P\setminus\{p\}\}$. If $|\mathcal{L}|\leq deg(p)+deg(q)+\nu_{2}(P,\mathcal{L})-3$, then $\tau(P,\mathcal{L})\leq\nu_{2}(P,\mathcal{L})-1$.

4 Intersecting $r$-uniform linear systems with $\tau=\nu_{2}=r$

In this subsection, we give some properties of $r$-uniform linear systems that satisfies $\tau=\nu_{2}=r$ as well as a characterization of $4$-uniform linear systems with $\tau=\nu_{2}=4$.

Let $\mathbb{L}_{r}$ be the family of intersecting linear systems $(P,\mathcal{L})$ of rank $r$ that satisfies $\tau(P,\mathcal{L})=\nu_{2}(P,\mathcal{L})=r$, then we have the following lemma:

Lemma 4.1**.**

Each element of $\mathbb{L}_{r}$ is an $r$-uniform linear system.

Proof

Let consider $(P,\mathcal{L})\in\mathbb{L}_{r}$ and $l\in\mathcal{L}$ any line of $(P,\mathcal{L})$. It is clear that $T=\{p\in l:deg(p)\geq 2\}$ is a transversal of $(P,\mathcal{L})$. Hence $r=\tau(P,\mathcal{L})\leq|T|\leq r$, which implies that $|l|=r$, for all $l\in\mathcal{L}$. Moreover, $deg(p)\geq 2$, for all $p\in l$ and $l\in\mathcal{L}$. ∎

In [8] was proved the following:

Lemma 4.2**.**

[8]** Let $(P,\mathcal{L})$ be an $r$-uniform intersecting linear system then every edge of $(P,\mathcal{L})$ has at most one vertex of degree 2. Moreover $\Delta(P,\mathcal{L})\leq r$.

Lemma 4.3**.**

[8]** Let $(P,\mathcal{L})$ be an $r$-uniform intersecting linear system then

[TABLE]

Hence, by Theorem 2.1 and Lemma 4.3 it follows:

Corollary 4.1**.**

If $(P,\mathcal{L})\in\mathbb{L}_{r}$ then $3(r-1)+1\leq|\mathcal{L}|\leq r^{2}-r+1$.

In [2] was proved that the linear systems $(P,\mathcal{L})$ with $|\mathcal{L}|>\nu_{2}(P,\mathcal{L})$ and $\nu_{2}(P,\mathcal{L})\in\{2,3,4\}$ satisfy $\tau(P,\mathcal{L})\leq\nu_{2}(P,\mathcal{L})$; and when attain the equality, they are a special family of linear subsystems of the projective plane of order $3$, $\Pi_{3}$ (some of them 4-uniform intersecting linear systems) with transversal and $2$-packing numbers equal to $4$. Recall that a finite projective plane (or merely projective plane) is a linear system satisfying that any pair of points have a common line, any pair of lines have a common point and there exist four points in general position (there are not three collinear points). It is well known that, if $(P,\mathcal{L})$ is a projective plane, there exists a number $q\in\mathbb{N}$, called order of projective plane, such that every point (line, respectively) of $(P,\mathcal{L})$ is incident to exactly $q+1$ lines (points, respectively), and $(P,\mathcal{L})$ contains exactly $q^{2}+q+1$ points (lines, respectively). In addition to this, it is well known that projective planes of order $q$, denoted by $\Pi_{q}$, exist when $q$ is a power prime. For more information about the existence and the unicity of projective planes see, for instance, [3, 6].

Given a linear system $(P,\mathcal{L})$, a triangle $\mathcal{T}$ of $(P,\mathcal{L})$, is the linear subsystem of $(P,\mathcal{L})$ induced by three points in general position (non collinear) and the three lines induced by them. In [2] was defined $\mathcal{C}=(P_{\mathcal{C}},\mathcal{L}_{\mathcal{C}})$ to be the linear system obtained from $\Pi_{3}$ by deleting $\mathcal{T}$; also there was defined $\mathcal{C}_{4,4}$ to be the family of linear systems $(P,\mathcal{L})$ with $\nu_{2}(P,\mathcal{L})=4$, such that:

• i)

$\mathcal{C}$ is a linear subsystem of $(P,\mathcal{L})$; and

• ii)

$(P,\mathcal{L})$ is a linear subsystem of $\Pi_{3}$,

this is \mathcal{C}_{4,4}=\{(P,\mathcal{L}):\mathcal{C}\subseteq(P,\mathcal{L})\subseteq\Pi_{3}\mbox{ and \nu_{2}(P,\mathcal{L})=4}\}.

Hence, the authors proved the following:

Theorem 4.1**.**

[2]** Let $(P,\mathcal{L})$ be a linear system with $\nu_{2}(P,\mathcal{L})=4$. Then, $\tau(P,\mathcal{L})=\nu_{2}(P,\mathcal{L})=4$ if and only if $(P,\mathcal{L})\in\mathcal{C}_{4,4}$.

Now, consider the projective plane $\Pi_{3}$ and a triangle $\mathcal{T}$ of $\Pi_{3}$ (see $(a)$ of Figure 2). Define $\mathcal{\hat{C}}=(P_{\mathcal{C}},\mathcal{L}_{\mathcal{C}})$ to be the linear subsystem induced by $\mathcal{L}_{\mathcal{C}}=\mathcal{L}\setminus\mathcal{T}$ (see $(b)$ of Figure 2). The linear system $\mathcal{\hat{C}}=(P_{\mathcal{C}},\mathcal{L}_{\mathcal{C}})$ just defined has ten points and ten lines. Define $\mathcal{\hat{C}}_{4,4}$ to be the family of 4-uniform intersecting linear systems $(P,\mathcal{L})$ with $\nu_{2}(P,\mathcal{L})=4$, such that:

• i)

$\mathcal{\hat{C}}$ is a linear subsystem of $(P,\mathcal{L})$; and

• ii)

$(P,\mathcal{L})$ is a linear subsystem of $\Pi_{3}$,

It is clear that $\mathcal{\hat{C}}_{4,4}\subseteq\mathcal{C}_{4,4}$ and each linear system $(P,\mathcal{L})\in\mathcal{\hat{C}}_{4,4}$ is an 4-uniform intersecting linear system. Hence

Corollary 4.2**.**

$(P,\mathcal{L})\in\mathbb{L}_{4}$* if and only if $(P,\mathcal{L})\in\mathcal{\hat{C}}_{4,4}$.*