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The kind of object you're looking at is exactly an $(r, \lambda)$-design for $\lambda = 1$ in combinatorial design theory.
An $(r, \lambda)$-design is an ordered pair $(V, \mathcal{B})$ of finite sets, mathcal{B})$, where$\mathcal{B}$is a set collection of subsets of finite set$V$such that every element of$V$appears in exactly$r$elements of$\mathcal{B}$, and every pair of distinct elements of$V$appears in exactly$\lambda$elements of$\mathcal{B}$. Usually, the cardinalitys$\vert V \vert$and$\vert \mathcal{B} \vert$are written as$v$and$b$respectively. We call the elements of$V$points and those of$\mathcal{B}$blocks. To see the equivalence, call each card a point, and each symbol a block. The set$A =\lbrace a_1, \dots, a_n \rbrace$is the set$\mathcal{B}$of blocks here, and you say a point is contained in block$a_i$if the corresponding card has the symbol$a_i$on it. Then setting$r = k$and$\lambda = 1$, the above definition defines exactly what you described in the language of cards with symbols on them; every card has$k$symbols (= every point appears$r$times) and every pair of cards share exactly one symbol of$A$(= every pair of points appear exactly once in a block in$\mathcal{B}$). The question you asked can be understood as "What's the maximum number of points in a$(k,1)$-design when the number of blocks is$n$?" To answer this, the basic relation between parameters of a nontrivial$(r, \lambda)$-design is:$v \leq r(r-1)+1$with equality if$r-\lambda$is the order of a finite projective plane. The following might also be helpful if you ask the same kind of question by fixing some parameters:$b \geq v r^2/(r+\lambda(v-1))$,$c_i(r-\lambda+\lambda v - \lambda k_i) \leq (r-\lambda)(r-\lambda +\lambda v)$, (c_i is the size of the$i$th block or equivalently the number of cards that has have the symbol$i$). You can find more about$(r, \lambda)$-designs in chapter "(r, \lambda)-designs" by G.H.J. van Rees in the book "Handbook of Combinatorial Designs" edited by C.J. Colbourn and J. Dinitz. Edit: The correct definition should allow the same subset of$V$appearing more than once in$\mathcal{B}$, so$\mathcal{B}$shouldn't be a set but a collection. 2 added 10 characters in body The kind of object you're looking at is exactly an$(r, \lambda)$-design for$\lambda = 1$in combinatorial design theory. An$(r, \lambda)$-design is an ordered pair$(V, \mathcal{B})$of finite sets, where$\mathcal{B}$is a set of subsets of$V$such that every element of$V$appears in exactly$r$elements of$\mathcal{B}$, and every pair of distinct elements of$V$appears in exactly$\lambda$elements of$\mathcal{B}$. Usually, the cardinalitys$\vert V \vert$and$\vert \mathcal{B} \vert$are written as$v$and$b$respectively. We call the elements of$V$points and those of$\mathcal{B}$blocks. To see the equivalence, call each card a point, and each symbol a block. The set$A =\lbrace a_1, \dots, a_n \rbrace$is the set$\mathcal{B}$of blocks here, and you say a point is contained in block$a_i$if the corresponding card has the symbol$a_i$on it. Then setting$r = k$and$\lambda = 1$, the above definition defines exactly what you described in the language of cards with symbols on them; every card has$k$symbols (= every point appears$r$times) and every pair of cards share exactly one symbol of$A$(= every pair of points appear exactly once in a block in$\mathcal{B}$). The question you asked can be understood as "What's the maximum number of points in a$(k,1)$-design when the number of blocks is$n$?" To answer this, the basic relation between parameters of an a nontrivial$(r, \lambda)$-design is:$v \leq r(r-1)+1$with equality if$r-\lambda$is the order of a finite projective plane. The following might also be helpful if you ask the same kind of question by fixing some parameters:$b \geq v r^2/(r+\lambda(v-1))$,$c_i(r-\lambda+\lambda v - \lambda k_i) \leq (r-\lambda)(r-\lambda +\lambda v)$, (c_i is the size of the$i$th block or equivalently the number of cards that has the symbol$i$). You can find more about$(r, \lambda)$-designs in chapter "(r, \lambda)-designs" by G.H.J. van Rees in the book "Handbook of Combinatorial Designs" edited by C.J. Colbourn and J. Dinitz. 1 The kind of object you're looking at is exactly an$(r, \lambda)$-design for$\lambda = 1$in combinatorial design theory. An$(r, \lambda)$-design is an ordered pair$(V, \mathcal{B})$of finite sets, where$\mathcal{B}$is a set of subsets of$V$such that every element of$V$appears in exactly$r$elements of$\mathcal{B}$, and every pair of distinct elements of$V$appears in exactly$\lambda$elements of$\mathcal{B}$. Usually, the cardinalitys$\vert V \vert$and$\vert \mathcal{B} \vert$are written as$v$and$b$respectively. We call the elements of$V$points and those of$\mathcal{B}$blocks. To see the equivalence, call each card a point, and each symbol a block. The set$A =\lbrace a_1, \dots, a_n \rbrace$is the set$\mathcal{B}$of blocks here, and you say a point is contained in block$a_i$if the corresponding card has the symbol$a_i$on it. Then setting$r = k$and$\lambda = 1$, the above definition defines exactly what you described in the language of cards with symbols on them; every card has$k$symbols (= every point appears$r$times) and every pair of cards share exactly one symbol of$A$(= every pair of points appear exactly once in a block in$\mathcal{B}$). The question you asked can be understood as "What's the maximum number of points in a$(k,1)$-design when the number of blocks is$n$?" To answer this, the basic relation between parameters of an$(r, \lambda)$-design is:$v \leq r(r-1)+1$with equality if$r-\lambda$is the order of a finite projective plane. The following might also be helpful if you ask the same kind of question by fixing some parameters:$b \geq v r^2/(r+\lambda(v-1))$,$c_i(r-\lambda+\lambda v - \lambda k_i) \leq (r-\lambda)(r-\lambda +\lambda v)$, (c_i is the size of the$i$th block or equivalently the number of cards that has the symbol$i$). You can find more about$(r, \lambda)\$-designs in chapter "(r, \lambda)-designs" by G.H.J. van Rees in the book "Handbook of Combinatorial Designs" edited by C.J. Colbourn and J. Dinitz.