ul 2 00 1 A stronger form of the theorem constructing a rigid binary relation on any set

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 then there exists a relation R ⊆ A×A with the following property ∀ x 6=y ∃ {x}⊆A(x,y)⊆A cardA(x,y)≤κ ∀ f :A(x,y)→A f(x)=y f is not a homomorphism of R which implies that R is rigid. On every set A there is a rigid binary relation, i.e. such a relation R that there is no homomorphism < A,R >→< A,R > except the identity ([2],[3],[6]). Conjectures 1 and 2 below strengthen this theorem. Conjecture 1 ([4],[5]). If κ is an infinite cardinal number and card A ≤ 2 κ then there exists a relation R ⊆ A×A which satisfies the following condition (∗): (∗) ∀ x 6=y ∃ {x}⊆A(x,y)⊆A cardA(x,y)≤κ ∀ f :A(x,y)→A f(x)=y f is not a homomorphism of R. Mathematics Subject Classification (2000): 03E05, 08A35. Proposition 1 ([5]). If κ is an infinite cardinal number, R ⊆ A×A satisfies the condition (∗) and card à ≤ card A then there exists a relation R̃ ⊆ Ã × Ã which satisfies the condition (∗). Remark 1 ([4]). If R ⊆ A × A satisfies the condition (∗) then R is rigid. If κ is an infinite cardinal number and a relation R ⊆ A × A satisfies the condition (∗) then card A ≤ 2 κ . Theorem 1 ([5]). Conjecture 1 is valid for κ = ω. Conjecture 2 ([4],[5]). If κ 6= 0 is a limit cardinal number and card A ≤ 2sup{2 α:α∈Card,α<κ} then there exists a relation R ⊆ A × A which satisfies the following condition (∗∗): (∗∗) ∀ x 6=y ∃ {x}⊆A(x,y)⊆A cardA(x,y)<κ ∀ f :A(x,y)→A f(x)=y f is not a homomorphism of R. Proposition 2 ([5]). If κ 6= 0 is a limit cardinal number, R ⊆ A × A satisfies the condition (∗∗) and card à ≤ card A then there exists a relation R̃ ⊆ Ã × Ã which satisfies the condition (∗∗). Remark 2 ([4]). If R ⊆ A × A satisfies the condition (∗∗) then R is rigid. If κ 6= 0 is a limit cardinal number and a relation R ⊆ A×A satisfies the condition (∗∗) then card A ≤ 2sup{2 :α∈Card,α<κ}. Theorem 2 ([4]). Conjecture 2 is valid for κ = ω. In this note we prove some weaker form of Conjecture 1 which holds for all infinite cardinal numbers κ, this main result is stated in Theorem 3. Theorem 3. If κ is an infinite cardinal number and card A ≤ 2 then there exists a relation R ⊆ A× A which satisfies the condition (∗). Proof. It is known ([1],[2],[3]) that for each infinite cardinal number κ there exists a rigid symmetric relation R ⊆ κ×κ. Let Φ denote the family of all relations S ⊆ κ× κ which satisfy: