TY - JOUR
T1 - Strong failures of higher analogs of Hindman’s theorem
AU - Fernández-Bretón, David
AU - Rinot, Assaf
N1 - Publisher Copyright:
© 2017 American Mathematical Society.
PY - 2017/12
Y1 - 2017/12
N2 - This paper is dedicated to the memory of András Hajnal (1931-2016) Abstract. We show that various analogs of Hindman’s theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring c: R → Q, such that for every X ⊆ R with |X| = |R|, and every colour γ ∈ Q, there are two distinct elements x0, x1 of X for which c(x0 + x1) = γ. This forms a simultaneous generalization of a theorem of Hindman, Leader and Strauss and a theorem of Galvin and Shelah. Theorem 2. For every abelian group G, there exists a colouring c: G → Q such that for every uncountable X ⊆ G and every colour γ, for some large enough integer n, there are pairwise distinct elements x0,…, xn of X such that c(x0 + ・ ・ ・ + xn) = γ. In addition, it is consistent that the preceding statement remains valid even after enlarging the set of colours from Q to R. Theorem 3. Let _κ assert that for every abelian group G of cardinality κ, there exists a colouring c: G → G such that for every positive integer n, every X0,…,Xn ∈ [G]κ, and every γ ∈ G, there are x0 ∈ X0,…, xn ∈ Xn such that c(x0 + ・ ・ ・ + xn) = γ. Then (Formula Found) holds for unboundedly many uncountable cardinals κ, and it is consistent that (Formula Found) holds for all regular uncountable cardinals κ.
AB - This paper is dedicated to the memory of András Hajnal (1931-2016) Abstract. We show that various analogs of Hindman’s theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring c: R → Q, such that for every X ⊆ R with |X| = |R|, and every colour γ ∈ Q, there are two distinct elements x0, x1 of X for which c(x0 + x1) = γ. This forms a simultaneous generalization of a theorem of Hindman, Leader and Strauss and a theorem of Galvin and Shelah. Theorem 2. For every abelian group G, there exists a colouring c: G → Q such that for every uncountable X ⊆ G and every colour γ, for some large enough integer n, there are pairwise distinct elements x0,…, xn of X such that c(x0 + ・ ・ ・ + xn) = γ. In addition, it is consistent that the preceding statement remains valid even after enlarging the set of colours from Q to R. Theorem 3. Let _κ assert that for every abelian group G of cardinality κ, there exists a colouring c: G → G such that for every positive integer n, every X0,…,Xn ∈ [G]κ, and every γ ∈ G, there are x0 ∈ X0,…, xn ∈ Xn such that c(x0 + ・ ・ ・ + xn) = γ. Then (Formula Found) holds for unboundedly many uncountable cardinals κ, and it is consistent that (Formula Found) holds for all regular uncountable cardinals κ.
KW - Commutative cancellative semigroups
KW - Hindman’s theorem
KW - JÓnsson cardinal
KW - Strong coloring
UR - http://www.scopus.com/inward/record.url?scp=85032257766&partnerID=8YFLogxK
U2 - 10.1090/tran/7131
DO - 10.1090/tran/7131
M3 - Artículo
AN - SCOPUS:85032257766
SN - 0002-9947
VL - 369
SP - 8939
EP - 8966
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 12
ER -