My question arises from a construction I gave in my recent answer to a question of Alexander Pruss concerning large families of independent non-measurable sets of reals. In that argument, using the continuum hypothesis and the existence of a thick Kurepa tree $T$, I produced a family of $2^{\frak c}$ many Vitali sets $\{\ A_s\mid s\in[T]\ \}$, which was almost disjoint in the sense that $A_s\cap A_t$ was countable whenever $s\neq t$. The only aspect of the Vitali relation that was used in the construction was that the Vitali equivalence classes (equivalence under rational translation) are countably infinite. Thus, the construction proves:

Theorem. If the CH holds and there is a thick Kurepa tree, then for every partition of $\mathbb{R}$ into countably infinite sets, there is an almost-disjoint family of selectors of size $2^{\frak c}$.

By almost-disjoint here, I mean that any two distinct elements of the family have countable intersection; by selector, I mean that each set in the family has exactly one element from each equivalence class; and by a partition into countably infinite sets, I mean that we have an equivalence relation on $\mathbb{R}$ with every equivalence class countably infinite. To prove the theorem, simply label the nodes on the $\alpha^{th}$ level of $T$ with distinct members of the $\alpha^{th}$ equivalence class in the partition. Being thick, the tree has $2^{\frak c}$ many branches, each of which provides a selector, and any two such selectors can agree only up to some countable height in the tree, where those branches separate.

My question is whether I really needed those set-theoretic assumptions in order to make the conclusion.

Question. How much can one weaken the hypotheses of the theorem and still prove the conclusion?

For example, can we drop the thick Kurepa tree assumption? Can we omit CH? Can we prove it in ZFC? Can one show the consistency with ZFC of a counterexample?

My question arises from a construction I gave in my recent answer to a question of Alexander Pruss concerning large families of independent non-measurable sets of reals. In that argument, using the continuum hypothesis and the existence of a thick Kurepa tree $T$, I produced a family of $2^{\frak c}$ many Vitali sets $\{\ A_s\mid s\in[T]\ \}$, which was almost disjoint in the sense that $A_s\cap A_t$ was countable whenever $s\neq t$. The only aspect of the Vitali relation that was used in the construction was that the Vitali equivalence classes (equivalence under rational translation) are countably infinite. Thus, the construction proves:

Theorem. If the CH holds and there is a thick Kurepa tree, then for every partition of $\mathbb{R}$ into countably infinite sets, there is an almost-disjoint family of selectors of size $2^{\frak c}$.

By almost-disjoint here, I mean that any two distinct elements of the family have countable intersection; by selector, I mean that each set in the family has exactly one element from each equivalence class; and by a partition into countably infinite sets, I mean that we have an equivalence relation on $\mathbb{R}$ with every equivalence class countably infinite. To prove the theorem, simply label the nodes on the $\alpha^{th}$ level of $T$ with distinct members of the $\alpha^{th}$ equivalence class in the partition. Being thick, the tree has $2^{\frak c}$ many branches, each of which provides a selector, and any two such selectors can agree only up to some countable height in the tree, where those branches separate.

My question is whether I really needed those set-theoretic assumptions in order to make the conclusion.

Question. How much can one weaken the hypotheses of the theorem and still prove the conclusion?

For example, can we drop the thick Kurepa tree assumption? Can we omit CH? Can we prove it in ZFC? Can one show the consistency with ZFC of a counterexample?

My question arises from a construction I gave in my recent answer to a question of Alexander Pruss concerning large families of independent non-measurable sets of reals. In that argument, using the continuum hypothesis and the existence of a thick Kurepa tree $T$, I produced a family of $2^{\frak c}$ many Vitali sets $\{\ A_s\mid s\in[T]\ \}$, which was almost disjoint in the sense that $A_s\cap A_t$ was countable whenever $s\neq t$. The only aspect of the Vitali relation that was used in the construction was that the Vitali equivalence classes (equivalence under rational translation) are countable. Thus, the construction proves:

Theorem. If the CH holds and there is a thick Kurepa tree, then for every partition of $\mathbb{R}$ into countable sets, there is an almost-disjoint family of selectors of size $2^{\frak c}$.

By almost-disjoint here, I mean that any two distinct elements of the family have countable intersection; and by selector, I mean that each set in the family has exactly one element from each equivalence class. To prove the theorem, simply label the nodes on the $\alpha^{th}$ level of $T$ with distinct members of the $\alpha^{th}$ equivalence class in the partition. Being thick, the tree has $2^{\frak c}$ many branches, each of which provides a selector, and any two such selectors can agree only up to some countable height in the tree, where those branches separate.

My question is whether I really needed those set-theoretic assumptions in order to make the conclusion.

Question. How much can one weaken the hypotheses of the theorem and still prove the conclusion?

For example, can we drop the thick Kurepa tree assumption? Can we omit CH? Can we prove it in ZFC? Can one show the consistency with ZFC of a counterexample?

My question arises from a construction I gave in my recent answer to a question of Alexander Pruss concerning large families of independent non-measurable sets of reals. In that argument, using the continuum hypothesis and the existence of a thick Kurepa tree $T$, I produced a family of $2^{\frak c}$ many Vitali sets $\{\ A_s\mid s\in[T]\ \}$, which was almost disjoint in the sense that $A_s\cap A_t$ was countable whenever $s\neq t$. The only aspect of the Vitali relation that was used in the construction was that the Vitali equivalence classes (equivalence under rational translation) are countably infinite. Thus, the construction proves:

Theorem. If the CH holds and there is a thick Kurepa tree, then for every partition of $\mathbb{R}$ into countably infinite sets, there is an almost-disjoint family of selectors of size $2^{\frak c}$.

By almost-disjoint here, I mean that any two distinct elements of the family have countable intersection; by selector, I mean that each set in the family has exactly one element from each equivalence class; and by a partition into countably infinite sets, I mean that we have an equivalence relation on $\mathbb{R}$ with every equivalence class countably infinite. To prove the theorem, simply label the nodes on the $\alpha^{th}$ level of $T$ with distinct members of the $\alpha^{th}$ equivalence class in the partition. Being thick, the tree has $2^{\frak c}$ many branches, each of which provides a selector, and any two such selectors can agree only up to some countable height in the tree, where those branches separate.

My question is whether I really needed those set-theoretic assumptions in order to make the conclusion.

Question. How much can one weaken the hypotheses of the theorem and still prove the conclusion?

For example, can we drop the thick Kurepa tree assumption? Can we omit CH? Can we prove it in ZFC? Can one show the consistency with ZFC of a counterexample?