Macpherson in a survey of homogeneous structures, states that there are many $\aleph_0$-categorical structures which are not homogeneous. Here homogeneity is the ultrahomogeneity that is defined as every isomorphism between two finite substructures of a structure $M$ can be extended to an automorphism of $M$.

$ω$-homogeneity means for two $n$-tuples $\bar{a}$ and $\bar{b}$ with the same type, there is an automorphism $σ∈$$\rm Aut$$(M)$ such that $σ(\bar{a})=\bar{b}$, while ultrahomogeneity means for $n$-tuples $\bar{a}$ and $\bar{b}$ with the same quantifier-free type, there is an automorphism $σ∈$$\rm Aut$$(M)$ such that $σ(\bar{a})=\bar{b}$. Both atomic and countably saturated structures are $ω$-homogeneous. So $ℵ_0$-categorical structures are $ω$-homogeneous.

$ω$-homogeneity and ultrahomogeneity become equivalent if and only if the theory of a $ℵ_0$-categorical structure has quantifier elimination. Is there an example for $ℵ_0$-categorical structures without quantifier elimination?

Macpherson in a survey of homogeneous structures, states that there are many $\aleph_0$-categorical structures which are not homogeneous. Here homogeneity is the ultrahomogeneity that is defined as every isomorphism between two finite substructures of a structure $M$ can be extended to an automorphism of $M$. $\omega$-homogeneity means that any finite partial elementary mapping can be extended so that its domain includes any given element.

I am confused on this because it is well known that a $\aleph_0$-categorical structure is both atomic and countably saturated, and both atomic and countably saturated structures are $\omega$-homogeneous. This actually means that a $\aleph_0$-categorical structure is ultrahomogeneous. Where is wrong here?

Macpherson in a survey of homogeneous structures, states that there are many $\aleph_0$-categorical structures which are not homogeneous. Here homogeneity is the ultrahomogeneity that is defined as every isomorphism between two finite substructures of a structure $M$ can be extended to an automorphism of $M$.

I understand now that $ω$-homogeneity means for two $n$-tuples $\bar{a}$ and $\bar{b}$ with the same type, there is an automorphism $σ∈$$\rm Aut$$(M)$ such that $σ(\bar{a})=\bar{b}$, while ultrahomogeneity means for $n$-tuples $\bar{a}$ and $\bar{b}$ with the same quantifier-free type, there is an automorphism $σ∈$$\rm Aut$$(M)$ such that $σ(\bar{a})=\bar{b}$. Both atomic and countably saturated structures are $ω$-homogeneous. So $ℵ_0$-categorical structures are $ω$-homogeneous.

$ω$-homogeneity and ultrahomogeneity become equivalent if and only if the theory of a $ℵ_0$-categorical structure has quantifier elimination. Is there an example for $ℵ_0$-categorical structures without quantifier elimination?

Macpherson in a survey of homogeneous structures, states that there are many $\aleph_0$-categorical structures which are not homogeneous. Here homogeneity is the ultrahomogeneity that is defined as every isomorphism between two finite substructures of a structure $M$ can be extended to an automorphism of $M$.

$ω$-homogeneity means for two $n$-tuples $\bar{a}$ and $\bar{b}$ with the same type, there is an automorphism $σ∈$$\rm Aut$$(M)$ such that $σ(\bar{a})=\bar{b}$, while ultrahomogeneity means for $n$-tuples $\bar{a}$ and $\bar{b}$ with the same quantifier-free type, there is an automorphism $σ∈$$\rm Aut$$(M)$ such that $σ(\bar{a})=\bar{b}$. Both atomic and countably saturated structures are $ω$-homogeneous. So $ℵ_0$-categorical structures are $ω$-homogeneous.

$ω$-homogeneity and ultrahomogeneity become equivalent if and only if the theory of a $ℵ_0$-categorical structure has quantifier elimination. Is there an example for $ℵ_0$-categorical structures without quantifier elimination?

Macpherson in a survey of homogeneous structures, states that there are many $\aleph_0$-categorical structures which are not homogeneous. Here homogeneity is the ultrahomogeneity that is defined as every isomorphism between two finite substructures of a structure $M$ can be extended to an automorphism of $M$. $\omega$-homogeneity means that any finite partial elementary mapping can be extended so that its domain includes any given element.

I am confused on this because it is well known that a $\aleph_0$-categorical structure is both atomic and countably saturated, and both atomic and countably saturated structures are $\omega$-homogeneous. This actually means that a $\aleph_0$-categorical structure is ultrahomogeneous. Where is wrong here?

Macpherson in a survey of homogeneous structures, states that there are many $\aleph_0$-categorical structures which are not homogeneous. Here homogeneity is the ultrahomogeneity that is defined as every isomorphism between two finite substructures of a structure $M$ can be extended to an automorphism of $M$.

I understand now that $ω$-homogeneity means for two $n$-tuples $\bar{a}$ and $\bar{b}$ with the same type, there is an automorphism $σ∈$$\rm Aut$$(M)$ such that $σ(\bar{a})=\bar{b}$, while ultrahomogeneity means for $n$-tuples $\bar{a}$ and $\bar{b}$ with the same quantifier-free type, there is an automorphism $σ∈$$\rm Aut$$(M)$ such that $σ(\bar{a})=\bar{b}$. Both atomic and countably saturated structures are $ω$-homogeneous. So $ℵ_0$-categorical structures are $ω$-homogeneous.

$ω$-homogeneity and ultrahomogeneity become equivalent if and only if the theory of a $ℵ_0$-categorical structure has quantifier elimination. Is there an example for $ℵ_0$-categorical structures without quantifier elimination?