### A convex polygon as a discrete plane curve.

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Our short note gives the affirmative answer to one of Fishburn’s questions.

We derive new upper bounds for the densities of measurable sets in ${\mathbb{R}}^{n}$ which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions $2,\cdots ,24$. This gives new lower bounds for the measurable chromatic number in dimensions $3,\cdots ,24$. We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg,...

We study the maximum possible number $f(k,l)$ of intersections of the boundaries of a simple $k$-gon with a simple $l$-gon in the plane for $k,l\ge 3$. To determine the number $f(k,l)$ is quite easy and known when $k$ or $l$ is even but still remains open for $k$ and $l$ both odd. We improve (for $k\le l$) the easy upper bound $kl-l$ to $kl-\u2308k/6\u2309-l$ and obtain exact bounds for $k=5$$(f...$

We prove a “Tverberg type” multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.