Combinatorial set theory and cardinal function inequalities

Hodel, R. E.

doi:10.1090/S0002-9939-1991-1039531-7

Combinatorial set theory and cardinal function inequalities
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Proc. Amer. Math. Soc. 111 (1991), 567-575 Request permission

Abstract:

Three theorems of combinatorial set theory are proven. From the first we obtain the de Groot inequality $\left | X \right | \leq {2^{hL(X)}}$, the Ginsburg-Woods inequality $\left | X \right | \leq {2^{e(X)\Delta (X)}}$, the Erdös-Rado Partition Theorem for $n = 2$, and set-theoretic versions of the Hajnal-Juhász inequalities $\left | X \right | \leq {2^{c(X)\chi (X)}}$ and $\left | X \right | \leq {2^{s(X)\psi (X)}}$. From the second we obtain a generalization of the Arhangel’skiĭ inequality $\left | X \right | \leq {2^{L(X)\chi (X)}}$. From the third we obtain the Charlesworth inequality $n(X) \leq psw{(X)^{L(X)}}$ and a generalization of the Burke-Hodel inequality $\left | {K(X)} \right | \leq {2^{e(X)psw(X)}}$.

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