Papers for

combinatorial system designers

Papers whose findings have a practical use for this group, as judged from the abstract. Open a paper to read what it means in practice.

Oriented matroids counterexamples break longstanding combinatorics conjectures

Neither simpliciality nor mutation connectivity: conjectures of Las Vergnas and Cordovil-Las Vergnas fail

Abstract: We construct a simple rank-$7$ oriented matroid on $24$ elements with no simplicial tope, disproving the Las Vergnas simplex conjecture from 1980. We then show that the mutation graph on uniform oriented matroids is disconnected for infinitely many values of rank $r$ and ground-set size $n$, disproving the Cordovil--Las Vergnas conjecture from 1988.

Mon 14 SeptDiscrete Mathematics
The gist
Some old ideas in math thought every special shape called an oriented matroid had a simple kind of face, but this paper shows a counterexample where none exist. It also shows that certain transformations between these shapes can't always connect all shapes, disproving another guess from decades ago. The authors built specific examples proving these old beliefs wrong.
Open → 2609.15850v1