On binary relations without non-identical endomorphisms

Summary.

On every set A there is a rigid binary relation, i.e. a relation R such that there is no homomorphism $\langle A,R \rangle \to \langle A,R \rangle$ except the identity (Vopěnka et al. [1965]). We state two conjectures (cf. Tyszka [1994]) which strengthen this theorem, among them¶¶Conjecture 1. If $ \kappa $ is an infinite cardinal number and card $ A \leq 2^{2^{\kappa}} $ , then there exists a relation $ R \subseteq A \times A $ which satisfies¶¶$\forall^{x,y \in A}_{x \neq y}\ \exists^{\{x\} \subseteq A(x,y) \subseteq A}_{{\rm card} A(x,y) \leq \kappa}: \forall^{f:A(x,y) \rightarrow A}_{f(x)=y}$ f is not a homomorphism of R.¶We prove Conjecture 1 for $\kappa = \omega$, the first infinite cardinal number.