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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