Let $X$ be a nonvoid set of cardinal $\alpha$. Let $\cong$ be an equivalence relation on $X$. Let $\beta$ be the cardinal of the set \
(1)~~~~ $ D = \{ \, ( x, y ) \in X \times X ~|~ x \ncong y \, \} $ \
Let any family $( x_i )_{i \in I}$, with $I$ a nonvoid index set, and $x_i \in X$, for $i \in I$. We call such a family a {\it chain}chain, iff \
(2)~~~~ $ x_i \ncong x_j,~~ i, j \in I,~ i \neq j $ \
We denote by \
(3)~~~~ $ \kappa $ \
the smallest cardinal which is at least as large as the cardinal of any index set $I$ of a chain (2). \
Clearly, we shall have \
(4)~~~~ $ car ( X / \cong ) = \kappa $ \ $ car ( X / {\cong} ) = \kappa $
{\bf Problem 1} \Problem 1
Find, in terms of the cardinals $\alpha, \beta$, the cardinal $\kappa$. \
{\bf Problem 2} \Problem 2
Given the cardinal $\alpha$, and given an upper bound \
(5)~~~~ $ \beta \leq \gamma $ \
find, in terms of the cardinals $\alpha, \beta, \gamma$, an upper bound for the cardinal $\kappa$. \
{\bf Problem 3} \Problem 3
Given the cardinal $\alpha$, and given a lower bound \
(6)~~~~ $ \beta \geq \gamma $ \
find, in terms of the cardinals $\alpha, \beta, \gamma$, a lower bound for the cardinal $\kappa$. \ \
