More on lower bounds for partitioning α-large sets

AbstractContinuing the earlier research from [T. Bigorajska, H. Kotlarski, Partitioning α-large sets: some lower bounds, Trans. Amer. Math. Soc. 358 (11) (2006) 4981–5001] we show that for the price of multiplying the number of parts by 3 we may construct partitions all of whose homogeneous sets are much smaller than in [T. Bigorajska, H. Kotlarski, Partitioning α-large sets: some lower bounds, Trans. Amer. Math. Soc. 358 (11) (2006) 4981–5001]. We also show that the Paris–Harrington independent statement remains unprovable if the number of colors is restricted to 2, in fact, the statement ∀a,b∃dd→∗(a)2b+2 is unprovable in IΣb. Other results concern some lower bounds for partitions of pairs