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Karasev [16] conjectured that for every set of r blue lines, r green lines, and r red lines in the plane, there exists a partition of them into r colorful triples whose induced triangles intersect. We disprove this conjecture for every r and extend the counterexamples to higher dimensions.

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Correspondence to P. SOBERÓN.

Additional information

This project was done as part of the 2021 New York Discrete Math REU, funded by NSF grant DMS 2051026.

Carvalho's research was supported by the KINSC Summer Scholar program by Haverford College.

Soberón's research is supported by NSF grant DMS 2054419 and a PSC-CUNY TRADB52 award.

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Key words and phrases

  • Tverberg's theorem
  • hyperplane arrangement
  • combinatorial geometry

Mathematics Subject Classification

  • 52A37
  • 52C35