Putting a diamond inside the square

Abstract. By a 35-year-old theorem of Shelah, ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }+\mathrm{♢}({\lambda }^{+})$ does not imply square-with-built-in-diamond_lambda for regular uncountable cardinals $\lambda$.

Here, it is proved that ${\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}}_{\lambda }+\mathrm{♢}({\lambda }^{+})$ is equivalent to square-with-built-in-diamond_lambda for every singular cardinal $\lambda$.

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Citation information:

A. Rinot, Putting a diamond inside the square, Bull. Lond. Math. Soc., 47(3): 436-442, 2015.