Combinatorial set theory and cardinal function inequalities

Abstract

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