Chen 2018 cardinal characteristics continuum partitions

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William Chen, Shimon Garti, Thilo Weinert, Cardinal characteristics of the continuum and partitions, Israel Journal of Mathematics 235 (2020), no. 1, 13--38, DOI 10.1007/s11856-019-1942-y (online 4 November 2019). The version read is arXiv:1801.00238v1 (31 December 2017, 26 pages), the only arXiv version, whose record carries this journal reference; the theorem numbers here are v1's. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1801.00238), every other right reserved.

The paper derives negative partition relations from hypotheses far weaker than the continuum hypothesis, namely that certain cardinal characteristics at a regular kappa equal kappa^+ or that the stick principle holds. Theorem 2.9 proves that if kappa is regular and lambda = kappa^+ = d_kappa then lambda^2 does not arrow (lambdakappa, 4)^2, with Corollary 2.10 extending the failure to all alpha < lambda^2 * kappa; specialized to kappa = omega this is exactly the relation omega_1^2 does not arrow (omega_1omega, 4)^2 under d = aleph_1, and it answers a question of Jean Larson affirmatively. Parallel results run through the stick principle: Theorem 3.1 gives kappa^+ does not arrow (kappa^+, (kappa:2))^2 from stick(kappa) = kappa^+, while Theorem 4.3 and Corollary 4.5 give lambda^2 does not arrow (lambdakappa,4)^2 and, for b = aleph_1 = stick, alpha does not arrow (omega_1omega,4)^2 for all alpha < omega_1^2 * omega. Polarized relations are also treated (Theorems 2.2, 2.3, 2.5 and 2.7, of which 2.2, 2.5 and 2.7 answer problems of Garti and Shelah), and Theorem 5.1 shows the existence of a Luzin set yields a coloring witnessing omega_1 does not arrow [omega_1]^2_omega with no rainbow triangle, weakening Shelah's CH hypothesis for a question of Erdős and Hajnal. The methods are diagonal constructions guided by scales, reaping and dominating families, and Luzin-set genericity. For Problem 1171, the paper supplies the conditional negative relation omega_1^2 does not arrow (omega_1*omega,4)^2 under d = aleph_1 and surveys the surrounding partition relations.

Source: https://arxiv.org/abs/1801.00238.

Bears on. #1171

Results to transcribe.

• Theorem 2.9: If kappa is regular and lambda = kappa^+ = d_kappa then lambda^2 does not arrow (lambdakappa, 4)^2; for kappa = omega this is omega_1^2 not arrow (omega_1omega,4)^2 under d = aleph_1.
• Corollary 2.10: Under the same hypothesis, alpha does not arrow (lambda*kappa,4)^2 for all alpha < lambda^2 * kappa.
• Theorem 3.1: For regular kappa, stick(kappa) = kappa^+ implies kappa^+ does not arrow (kappa^+, (kappa:2))^2.
• Theorem 4.3 and Corollary 4.5: If lambda = kappa^+ = b_kappa = stick(kappa) then lambda^2 does not arrow (lambdakappa,4)^2; in particular b = aleph_1 = stick gives alpha not arrow (omega_1omega,4)^2 for all alpha < omega_1^2 * omega.
• Theorem 2.2: If b = d then (d over omega) does not arrow the square-bracket polarized relation [b over omega]^{1,1}_{aleph_0}; this answers Problems 3.6 and 3.10 of Garti-Shelah negatively and solves their Problem 3.19.
• Theorem 5.1: The existence of a Luzin set yields a coloring witnessing omega_1 not arrow [omega_1]^2_omega with no triangle of three distinct colors.

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