We are writing a paper in which we need to use the following theorem to prove that a certain poset satisfies c.c.c. As far as I remember I learned this theorem from András Hajnal.

Theorem: If $\eta<\omega_1$ is an ordinal, and $f:\omega_1\to \mathcal P(\omega_1)$ is a mapping such that $tp(f(\alpha))<\eta$ for each $\alpha<\omega_1$, then there is an uncountable $f$-free set $A$ (i.e. $\alpha\notin f(\beta)$ for each $\{\alpha,\beta\}\in [\omega_1]^2$).

(Lázár considered and solved the special case $\eta=\omega$ in 1936).

I can prove this Theorem, but I can not find a reference. Could you help me?

We are writing a paper in which we need to use the following theorem to prove that a certain poset satisfies c.c.c. As far as I remember I learned this theorem from András Hajnal.

Theorem 1: If $\eta<\omega_1$ is an ordinal, and $f:\omega_1\to \mathcal P(\omega_1)$ is a mapping such that $tp(f(\alpha))<\eta$ for each $\alpha<\omega_1$, then there is an uncountable $f$-free set $A$ (i.e. $\alpha\notin f(\beta)$ for each $\{\alpha,\beta\}\in [A]^2$).

(Lázár considered and solved the special case $\eta=\omega$ in 1936).

I can prove this Theorem, but I can not find a reference. Could you help me?

UPDATE. Fodor proved the following strengthening of Lázár's result:

Theorem 2: If $f:\omega_1\to [\omega_1]^{<\omega}$ is a mapping , then there is an partition $\{A_n:n<\omega \}$ of $\omega_1$ into $f$-free sets.

What about the following common generalization of Theorem 1 and 2?

Problem: If $\eta<\omega_1$ is an ordinal, and $f:\omega_1\to \mathcal P(\omega_1)$ is a mapping such that $tp(f(\alpha))<\eta$ for each $\alpha<\omega_1$, then there is an partition $\{A_n:n<\omega \}$ of $\omega_1$ into $f$-free sets.

The proofs I know for Theorem 1 and Theorem 2 do not work here.