An analogue to infinitery Hales-Jewett theorem

Abstract: In a recent work, N. Hindman, D. Strauss and L. Zamboni have shown that the Hales-Jewett theorem can be combined with a sufficiently well behaved homomorphisms. In this paper we will show that those combined extensions can be made if we replace the alphabet by an increasing sequence of alphabets, infact it holds for some Ramsey theoretic small sets. To obtained this we achieved some interesting configurations.