On a conjecture by Kalai
  We show that monomial ideals generated in degree two satisfy a conjecture by Eisenbud, Green and Harris. In particular we give a partial answer to a conjecture of Kalai by proving that $h$-vectors of flag Cohen-Macaulay simplicial complexes are $h$-vectors of Cohen-Macaulay balanced simplicial complexes. 
Giulio Caviglia, Alexandru Constantinescu and Matteo Varbaro
math.AC math.CO