AFFINE and PROJECTIVE SPACES

Let $\ X\ $ be an affine or a projective space (of arbitrary dimension), say, over a field $\ F\ $. Then every straight line $\ L\ $ of $\ X\ $ has cardinality equal to $\ \kappa:=|F|\ $ or $\ \kappa:=|F|+1\ $ respectively (these affine and projective cardinalities differ by $\ 1\ $ iff $\ F\ $ is finite). The set $\ E\subseteq \binom X\kappa\ $ of all straight lines in $\ X\ $ is linear and provides a solution to the OP question, when $\ |X|=|\omega|.$

-- Thus, there are two solutions for every finite field $\ F\ $, i.e. there are solutions for each $\ \kappa=p^n\ $ and each $\ \kappa=p^n+1\ $ where $\ p\ $ is an arbitrary prime and $\ n\in\mathbb N\ $ is an arbitrary natural number.

There are also two solutions for each field $\ F\ $ such that $\ |F|=|X|=|\omega|,\ $ -- for instance, let $\ F=\mathbb Q\ $ be the field of rational numbers, etc. Then $\ \kappa=|\omega|.$

These finite algebraic examples of $\ \kappa\ $ start with all

$$ 2\ \le\ \kappa\ \le\ 14 $$

followed by $$ 16 \le\ \kappa\ \le\ 20 $$ followed by $$ 23 \le\ \kappa\ \le\ 33 $$

etc. Some of these $\ \kappa\ $ are affine, some are projective, and some -- but very few -- are both, e.g.

$$ 2^2=3+1\qquad 5=2^2+1\qquad 2^3=7+1\qquad 3^2=2^3+1\qquad 17=2^4+1 $$

etc., in particular, Fermat primes are both; and so $\ \kappa=M+1\ $ where $\ M\ $ is a Mersenne prime. However, as it is well known, there is only one solution to $\ |p^n-q^m|=1,\ $ where $\ p\ $ and $\ q\ $ are primes, and natural numbers $\ n\ $ and $\ m\ $ are both greater than $1$. Thus, for $\ \kappa>17\ $ only the Fermat and Mersenne cases remain.

AFFINE and PROJECTIVE SPACES

Let $\ X\ $ be an affine or a projective space (of arbitrary dimension), say, over a field $\ F\ $. Then every straight line $\ L\ $ of $\ X\ $ has cardinality equal to $\ \kappa:=|F|\ $ or $\ \kappa:=|F|+1\ $ respectively (these affine and projective cardinalities differ by $\ 1\ $ iff $\ F\ $ is finite). The set $\ E\subseteq \binom X\kappa\ $ of all straight lines in $\ X\ $ is linear and provides a solution to the OP question, when $\ |X|=|\omega|.$

-- Thus, there are two solutions for every finite field $\ F\ $, i.e. there are solutions for each $\ \kappa=p^n\ $ and each $\ \kappa=p^n+1\ $ where $\ p\ $ is an arbitrary prime and $\ n\in\mathbb N\ $ is an arbitrary natural number.

There are also two solutions for each field $\ F\ $ such that $\ |F|=|X|=|\omega|,\ $ -- for instance, let $\ F=\mathbb Q\ $ be the field of rational numbers, etc. Then $\ \kappa=|\omega|.$

These finite algebraic examples of $\ \kappa\ $ start with all

$$ 2\ \le\ \kappa\ \le\ 14 $$

followed by $$ 16 \le\ \kappa\ \le\ 20 $$ followed by $$ 23 \le\ \kappa\ \le\ 33 $$

etc. Some of these $\ \kappa\ $ are affine, some are projective, and some -- but very few -- are both, e.g.

$$ 2^2=3+1\qquad 5=2^2+1\qquad 2^3=7+1\qquad 3^2=2^3+1\qquad 17=2^4+1 $$

etc., in particular, Fermat primes are both; and so $\ \kappa=M+1\ $ where $\ M\ $ is a Mersenne prime. However, as it is well known, there is only one solution to $\ |p^n-q^m|=1,\ $ where $\ p\ $ and $\ q\ $ are primes, and natural numbers $\ n\ $ and $\ m\ $ are both greater than $1$ (this a special case of the respective Tijdeman's theorem). Thus, for $\ \kappa>17\ $ only the Fermat and Mersenne cases remain.

AFFINE and PROJECTIVE SPACES

Let $\ X\ $ be an affine or a projective space (of arbitrary dimension), say, over a field $\ F\ $. Then every straight line $\ L\ $ of $\ X\ $ has cardinality equal to $\ \kappa:=|F|\ $ or $\ \kappa:=|F|+1\ $ respectively (these affine and projective cardinalities differ by $\ 1\ $ iff $\ F\ $ is finite). The set $\ E\subseteq \binom X\kappa\ $ of all straight lines in $\ X\ $ is linear and provides a solution to the OP question, when $\ |X|=|\omega|.$

-- Thus, there are two solutions for every finite field $\ F\ $, i.e. there are solutions for each $\ \kappa=p^n\ $ and each $\ \kappa=p^n+1\ $ where $\ p\ $ is an arbitrary prime and $\ n\in\mathbb N\ $ is an arbitrary natural number.

There are also two solutions for each field $\ F\ $ such that $\ |F|=|X|=|\omega|,\ $ -- for instance, let $\ F=\mathbb Q\ $ be the field of rational numbers, etc. Then $\ \kappa=|\omega|.$

These finite algebraic examples of $\ \kappa\ $ start with all

$$ 2\ \le\ \kappa\ \le\ 14 $$

followed by $$ 16 \le\ \kappa\ \le\ 20 $$ followed by $$ 23 \le\ \kappa\ \le\ 33 $$

etc. Some of these $\ \kappa\ $ are affine, some are projective, and some -- but very few -- are both, e.g.

$$ 2^2=3+1\qquad 5=2^2+1\qquad 2^3=7+1\qquad 3^2=2^3+1\qquad 17=2^4+1 $$

etc., in particular, Fermat primes are both. However, as it is well known, there is only one solution to $\ |p^n-q^m|=1,\ $ where $\ p\ $ and $\ q\ $ are primes, and natural numbers $\ n\ $ and $\ m\ $ are both greater than $1$.

AFFINE and PROJECTIVE SPACES

Let $\ X\ $ be an affine or a projective space (of arbitrary dimension), say, over a field $\ F\ $. Then every straight line $\ L\ $ of $\ X\ $ has cardinality equal to $\ \kappa:=|F|\ $ or $\ \kappa:=|F|+1\ $ respectively (these affine and projective cardinalities differ by $\ 1\ $ iff $\ F\ $ is finite). The set $\ E\subseteq \binom X\kappa\ $ of all straight lines in $\ X\ $ is linear and provides a solution to the OP question, when $\ |X|=|\omega|.$

-- Thus, there are two solutions for every finite field $\ F\ $, i.e. there are solutions for each $\ \kappa=p^n\ $ and each $\ \kappa=p^n+1\ $ where $\ p\ $ is an arbitrary prime and $\ n\in\mathbb N\ $ is an arbitrary natural number.

There are also two solutions for each field $\ F\ $ such that $\ |F|=|X|=|\omega|,\ $ -- for instance, let $\ F=\mathbb Q\ $ be the field of rational numbers, etc. Then $\ \kappa=|\omega|.$

These finite algebraic examples of $\ \kappa\ $ start with all

$$ 2\ \le\ \kappa\ \le\ 14 $$

followed by $$ 16 \le\ \kappa\ \le\ 20 $$ followed by $$ 23 \le\ \kappa\ \le\ 33 $$

etc. Some of these $\ \kappa\ $ are affine, some are projective, and some -- but very few -- are both, e.g.

$$ 2^2=3+1\qquad 5=2^2+1\qquad 2^3=7+1\qquad 3^2=2^3+1\qquad 17=2^4+1 $$

etc., in particular, Fermat primes are both; and so $\ \kappa=M+1\ $ where $\ M\ $ is a Mersenne prime. However, as it is well known, there is only one solution to $\ |p^n-q^m|=1,\ $ where $\ p\ $ and $\ q\ $ are primes, and natural numbers $\ n\ $ and $\ m\ $ are both greater than $1$. Thus, for $\ \kappa>17\ $ only the Fermat and Mersenne cases remain.