How much sweetness is there in the universe?

We continue investigations of forcing notions with strong ccc properties introducing new methods of building sweet forcing notions. We also show that quotients of topologically sweet forcing notions over Cohen reals are topologically sweet while the quotients over random reals do not have to be such. One of the main ingredients of the construction of the model for all projective sets of reals have the Baire property presented in Shelah [8, §7] was a strong ccc property of forcing notions called sweetness. This property is preserved in amalgamations and also in compositions with the Hechler forcing notion D and the Universal Meager forcing UM (see [8, §7]; a full explanation of how this is applied can be found in [3]). Stern [10] considered a slightly weaker property, topological sweetness, which is also preserved in amalgamations and compositions with D and UM. We further investigated the sweet properties of forcing notions in [5, §4], where we introduced a new property called iterable sweetness and we showed how one can build sweet forcing notions. New examples of iterably sweet forcing notions can be used in constructions like [8, §7], [7], but it could be that there is no need for this — the old forcing notions could be adding generic objects for all of them. In [6] we proved that this is exactly what happens with the natural examples of sweet forcing notions determined by the universality parameters as in [5, §2.3]: a sequence Cohen real — dominating real — Cohen real produces generic filters for many of them. In the present paper we show that sweetness is not so rare after all and we give more constructions of sweet forcing notions. In the first section we present a new method of building sweet forcing notions and we give our first example: a forcing notion Q T sc associated with scattered subtrees of 2 < ω. We do not know if the iterations of " old " forcing notions add generic objects for Q T sc , but in Proposition 1.8 we present an indication that this does not happen. In the second section we use our method to introduce two large families of sweet forcing notions, in some sense generalizing the known examples from [5]. This time we manage to show that some of our forcing notions are really new by showing that we have too many different examples (in Theorems …