### Consistently there is no non trivial ccc forcing notion with the Sacks or Laver property

by Shelah. [Sh:723]

Combinatorica, 2001

The result in the title answers a problem of Boban
Velickovic. A definable version of it (that is for Souslin forcing
notions) has been answered in [Sh 480], and our proof follows
it. Independently Velickovic proved this consistency, following
[Sh 480] and some of his works, proving it from PFA and from OCA. We
prove that moreover, consistently there is no ccc forcing with the
Laver property. Note that if cov(meagre)=continuum (which follows
e.g. from PFA) then there is a (non principal) Ramsey ultrafilter on
omega hence a forcing notion with the Laver property. So the
results are incomparable.

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