A stronger form of the theorem constructing a rigid binary relation on any set

A stronger form of the theorem constructing a rigid binary relation on any set Apoloniusz Tyszka Summary. On every set A there is a rigid binary relation i.e. such a relation R ⊆ A × A that there is no homomorphism A, R → A, R except the identity (Vopěnka et al. [1965]). We prove that for each infinite cardinal number κ if card A ≤ 2 κ , then there exists a relation R ⊆ A × A with the following property: ∀x ∈ A ∃ {x} ⊆ A(x) ⊆ A card A(x) ≤ κ ∀ f : A(x) → A f = id A(x) f is not a homomorphism of R which implies that R is rigid. If a relation R ⊆ A×A has the above property, then card A ≤ 2 κ. On every set A there is a rigid binary relation, i.e. such a relation R ⊆ A × A that there is no homomorphism A, R → A, R except the identity ([2],[3] [4],[7]). Conjectures 1 and 2 below strengthen this theorem.