What is the tension required for a rope with a finite mass per unit length, hanging between two fixed equal altitude points under gravity, so that the rope is perfectly horizontal (without any "slack"). Assume other properties of rope such as elastic constant, total mass, length, mass per unit length to be finite. And mass per unit length is uniform every point.
Just for your information, let me start by saying that the form of a rope hanging between two, say, equally-hight (exactly) vertical sticks, is a catenary (just as a rope hanging between two points that are not at equal height).
The rope can never be in an exactly horizontal form, no matter how great the tension. Gravity will always be present to introduce a bend in the rope.
As you said in your question, the properties of the rope such as elastic constant, total mass, length, mass per unit length are finite. This suggests we have to do with a real rope. For the rope to be perfectly horizontal we have to apply an infinite force to the rope, in the horizontal direction. Obviously, the rope will have snapped before reaching (the impossible) infinite force.
Even if the rope was an idealized one (unbreakable, with constant length), it wouldn't be possible because an infinite force doesn't exist. The rope would be exactly vertical in form (the horizontal deformation caused by gravity is overwhelmed by the infinite vertical force) in this case, but because of the simple fact that it is impossible to pull with infinite force on both ends of the rope, also this scenario would fail.