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(Long comment, not an answer to the question posed.) One way to generalise "general position" configurations of lines in the plane is to assign an orientation "clockwise" or "anticlockwise" to each s …

Note that it is possible to remove one or two points (maybe more) from Jukka's construction while retaining only 8 common lines. For example, removing the top left vertex of the array can be done like …

Let $q$ be a prime power and let $P$ be a projective plane of order $q$. It has $q^2+q+1$ points and $q^2+q+1$ lines. Each point lies on $q+1$ lines, and each line has $q+1$ points. Each pair of lines …

This paper has a reference to a positive answer to the first question.

I'll complement Christian's answer with an example in the other direction. Consider the polytope of $8\times 8$ symmetric doubly-stochastic matrices with 0 diagonal. The period of the Ehrhart quasipol …

It disturbs uniformity. What you have is a Markov chain and it converges to a distribution given by the Perron eigenvector of the transition matrix. To preserve uniformity you need that eigenvector t …

This is a standard problem in design theory. A Steiner system $S(t, k, v)$ is a pair $(X, B)$, where $X$ is a $v$-element set and $B$ is a set of $k$-subsets of $X$, called blocks, with the property …

(A comment rather than an answer.) Here is a plot of $a_n/n^5$ (red) and $b_n/n^5$ (blue). It might not go far enough to show the asymptotic behaviour, but a possibility is that $a_n$ and $b_n$ are as …