I want to see if it is possible to force the existence of a function 
$F:\aleph\_2 \times \aleph\_2\rightarrow \aleph\_1$ such that:


a) $F(a,b)=F(b,a)$, for all $a,b\in \aleph\_2$ and 


b) for all distinct $a,b$, the set $\{x|F(a,x)=F(b,x)\}$ is finite.


What is known:


1) Under CH, there is no such function. 


2) If such a function exists, let A be a subset of $\aleph\_2$ of size $\aleph\_1$. Consider the functions $F(a,\cdot)$ restricted to A. We have a family of $\aleph\_2$ many such functions from $\aleph\_1$ to $\aleph\_1$ and any two of them agree only on a finite set. That's a strongly almost disjoint family of size $\aleph\_2$. Baumgartner call this $A(\aleph\_1,\aleph\_2,\aleph\_1,\aleph\_0)$. It is consistent together with the negation of CH that either $A(\aleph\_1,\aleph\_2,\aleph\_1,\aleph\_0)$ or its negation hold. In particular, 
$\neg CH$ + "there is no such function" is consistent.


I want to see if anyone knows any result on the positive side.



Can we force the existence of this function?
First notice that we can separate the condition depending the relative position of $a,b$, and $x$ in (b): let $F\_0$ be such that for any $a,b$ there are finitely many $x<\min(a,b)$ with 
$F\_0(x,a)=F\_0(x,b)$, let $F\_1$ be likewise for $a<x<b$ and let $F\_2$ be for $a,b<x$. Then $F(a,b)=\langle F\_0(a,b),F\_1(a,b),F\_2(a,b)\rangle$ is as required. $F\_2$ obviously exists as we can have $F\_2(a,x)\neq F\_2(b,x)$ for $a\neq b<x$. It is well known that there is $F\_1:[c]^2\to\omega$ with $F\_2(a,x)\neq F\_2(x,b)$ for $a<x<b$ ($c$ is continuum). 
So there remains the task to force $F\_0$ and this can be done the way Miha suggested. 


Let $F:[\omega\_2]^2\to [\omega\_2]^{\leq\aleph\_0}$ be a Delta-function. We force with conditions $(s,f)$ where $s\in[\omega\_2]^{<\omega}$, $f:[\omega\_2]^2\to\omega$, if 
$x<a,b$ are in $s$, $f(x,a)=f(x,b)$, then $x\in F(a,b)$, and $(s',f')\leq (s,f)$ if $s'\supseteq s$, $f'\supseteq f$, and if $a,b\in s$, $x\in s'-s$, $x<a,b$, then 
$f'(x,a)\neq f'(x,b)$. 


The essential thing is to show ccc. Assume that we have $p\_\alpha=(s\cup s\_\alpha,f\_\alpha)\in P$, and $f\_\alpha|s$ are the same. By removing countably many $p\_\alpha$, we can get thet $s\_\alpha\cap F(a,b)=\emptyset$ for $a,b\in s$. By the Delta-function property, there are $\alpha<\beta$ such that if $x<a,b$, $x\in s$, $a\in s\_\alpha$, $b\in s\_\beta$, then $x\in F(a,b)$. Now we can extend $p\_\alpha$ and $p\_\beta$ to a common $p=(s',f')$ such that $f'(u,v)$ ($u\in s\_\alpha$, $v\in s\_\beta$) differ from each other and the range of $f\_\alpha, f\_\beta$. 


The consistency of the existence of a Delta function is in James E. Baumgartner and Saharon Shelah, Remarks on superatomic Boolean algebras, Ann. Pure Appl. Logic 33 (1987), no. 2, 109–129. 


