Linear syzygies, flag complexes, and regularity

@article{Constantinescu2015LinearSF,
  title={Linear syzygies, flag complexes, and regularity},
  author={Alexandru Constantinescu and Thomas Kahle and Matteo Varbaro},
  journal={Collectanea Mathematica},
  year={2015},
  volume={67},
  pages={357-362}
}
We show that for every $$r\in \mathbb {Z}_{>0}$$r∈Z>0 there exist monomial ideals generated in degree two, with linear syzygies, and regularity of the quotient equal to $$r$$r. For Gorenstein ideals we prove that the regularity of their quotients can not exceed four, thus showing that for $$d > 4$$d>4 every triangulation of a $$d$$d-manifold has a hollow square or simplex. We also show that for most monomial ideals generated in degree two and with linear syzygies the regularity is $$O(\log… 
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