More on lower bounds for partitioning -large sets

Continuing 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 ∃ d d → ∗ ( a ) 2 b + 2 is unprovable in I Σ b . Other results concern some lower bounds for partitions of pairs.