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Is the following conjecture true?

Conjecture: Let $M_1$ be a red map and let $M_2$ be a blue map drawn in general position on $S^n$, and let $M$ be their common refinement. There is a vertex $w$ of $M$, a blue vertex $v$ of $M_1$, a red vertex $u$ of $M_2$ and two faces $F,G$ of $M$ such that 1) $v,w \in F$ , 2) $u,w \in G$, and 3) $\dim F +\dim G =n$.

A simple (but perhaps not the most general) setting in which to ask this question is with regard to the red and blue maps coming from red and blue polyhedral fans associated to red and blue convex polytopes. The common refinement will be the fan obtained by taking all intersections of cones, one from the first fan and one from the second.

Motivation

For $n=2$ was proved by Paco Santos,Tamon Stephen and Hugh Thomas. They gave two proofs. One is based on an Euler characteristic argument, and the other applies a connectivity argument. Here is a link to the paper.

A positive answer to the conjecture in all dimensions will imply an upper bound of the form $nd$ to diameters of graphs of simple $d$-dimensional polytopes with $n$ facets. (Proving the conjecture for the simple case mentioned above would suffice.) This will be great as no polynomial upper bound is known. For more information see this post.

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What does the tag "polymath3" stand for? –  André Henriques Jul 13 '11 at 20:02
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A polymath project is a collaborative open way to study mathematical problem. It was proposed by Tim Gowers here gowers.wordpress.com/2009/01/27/… and Tim also ran polymath1 on his blog. gowers.wordpress.com/2009/01/30/… . (I dont remember how the name polymath was chosen.) Since then there were polymath1, polymath4, polymath5, and polymath 3 which were "full" projects and also polymath2, polymath7, Aaronson's philomath gilkalai.wordpress.com/2010/09/29/… –  Gil Kalai Jul 13 '11 at 20:18
    
(cont) polymath 2,7 and philomath spanned a single comment thread. Polymath3 was on my blog and dealt with the "polynomial Hirsch conjecture". The link to the starting post of polymath3 is here gilkalai.wordpress.com/2010/09/29/… . See also the polymath blog polymathprojects.org So far polymath1 was the only one that have led to a complete solution of the proposed problem. But the other projects also had some nice fruits. –  Gil Kalai Jul 13 '11 at 20:24
    
The tag polymath3 stands for MO questions related to the polymath3 project, There are similar tags polymath5 and polymath1 –  Gil Kalai Jul 13 '11 at 20:26

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