Question 6
Let $A_1,\ldots,A_n$ be $n>2$ points in space, no four coplanar. Join every pair of distinct points by a segment. Find the largest possible $n$ such that all these segments can be colored blue or red satisfying:
1. Each segment receives exactly one color.
2. At each $A_i$, at most four incident segments are blue.
3. For every red segment $A_iA_j$, there is a point $A_k$, different from both endpoints, for which both $A_kA_i$ and $A_kA_j$ are blue.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; the rubric is editorial. Importer multipliers: correct 1, incorrect 0, blank 0.
1. Each segment receives exactly one color.
2. At each $A_i$, at most four incident segments are blue.
3. For every red segment $A_iA_j$, there is a point $A_k$, different from both endpoints, for which both $A_kA_i$ and $A_kA_j$ are blue.
The supplied PDF does not state a time limit. This edition uses the user-approved 240 minutes per session, 480 minutes total for both sessions; this is an editorial convention. The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale. Essay grading is manual; the rubric is editorial. Importer multipliers: correct 1, incorrect 0, blank 0.