Lines in space

Task number: 4044

Determine bounds on the maximum number of intersections of a set of lines in the \( 3 \)-dimensional space, if you know that no three lines from this set can lie in a coomon plane. If three or more lines intersect at one point, we count such an intersection only once.

  • Hint

    For a upper bound, examine the properties of a graph whose vertices are straight lines and the edges are a pair of straight lines with a non-empty intersection.

    For a lower bound consider a square grid formed by \( n / 2 \) horizontal and \( n / 2 \) vertical lines in the plane (for \( n \) odd we omit one vertical line). Next, lift (tilt) the lines appropriately into the space so that the pairs of intersecting lines remain the same, but no three are in the same plane.

    An alternative way to get the lower bound is to look for lines determined by the cooling towers of the Temelín Nuclear Power Plant (one-piece hyperboloid).

Difficulty level: Moderate task
Solution require uncommon idea
Cs translation
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