[proofplan]
We first prove the two divisibility conditions by double-counting incidences between points and blocks, and then between unordered pairs of points and blocks. These counts show that every $2-(v,k,\lambda)$ design is admissible. For the converse, we invoke Wilson's 1972 asymptotic existence theorem for pairwise balanced designs as an external black-box result, of which the balanced incomplete block design case is the special case with all blocks of size $k$ and pair multiplicity $\lambda$. Applying that external theorem gives the required threshold $v_0(k,\lambda)$.
[/proofplan]
[step:Count blocks through one point to obtain the first congruence]
Assume that $(X,\mathcal{B})$ is a $2-(v,k,\lambda)$ design. We regard $\mathcal{B}$ as a finite multiset of $k$-element subsets of $X$, so repeated blocks, if present, are counted with their multiplicities. For a point $x\in X$, define the replication number
\begin{align*}
r_x := |\{B\in\mathcal{B}: x\in B\}|,
\end{align*}
where the cardinality counts blocks with multiplicity.
Count the set
\begin{align*}
I_x := \{(y,B)\in (X\setminus\{x\})\times\mathcal{B}: x\in B \text{ and } y\in B\}.
\end{align*}
For each $y\in X\setminus\{x\}$, the pair $\{x,y\}$ lies in exactly $\lambda$ blocks, so $|I_x|=\lambda(v-1)$. For each block $B\in\mathcal{B}$ containing $x$, there are exactly $k-1$ choices of $y\in B\setminus\{x\}$, so $|I_x|=r_x(k-1)$. Hence
\begin{align*}
r_x(k-1)=\lambda(v-1).
\end{align*}
Since $r_x$ is an integer, $k-1$ divides $\lambda(v-1)$, which is exactly
\begin{align*}
\lambda(v-1)\equiv 0\pmod{k-1}.
\end{align*}
[guided]
Fix a point $x\in X$. The goal is to extract an integer divisibility condition from the fact that every pair involving $x$ occurs exactly $\lambda$ times. Throughout the count, $\mathcal{B}$ is treated as a finite multiset of $k$-element subsets of $X$, so repeated blocks are counted according to their multiplicities. Define
\begin{align*}
r_x := |\{B\in\mathcal{B}: x\in B\}|,
\end{align*}
the number of blocks containing $x$, counted with multiplicity. This is an integer because $\mathcal{B}$ is a finite multiset of blocks.
We count the same finite set in two ways. Let
\begin{align*}
I_x := \{(y,B)\in (X\setminus\{x\})\times\mathcal{B}: x\in B \text{ and } y\in B\}.
\end{align*}
First choose $y$. There are $v-1$ choices of $y\in X\setminus\{x\}$, and for each such $y$, the defining property of a $2-(v,k,\lambda)$ design says that the two-element set $\{x,y\}$ lies in exactly $\lambda$ blocks. Therefore
\begin{align*}
|I_x|=\lambda(v-1).
\end{align*}
Now choose the block first. There are $r_x$ blocks containing $x$. Each such block has exactly $k$ points, and after fixing $x$ there are exactly $k-1$ remaining choices for $y$. Therefore
\begin{align*}
|I_x|=r_x(k-1).
\end{align*}
Equating the two counts gives
\begin{align*}
r_x(k-1)=\lambda(v-1).
\end{align*}
Since $r_x\in\mathbb{Z}$, the integer $k-1$ divides $\lambda(v-1)$, which is precisely
\begin{align*}
\lambda(v-1)\equiv 0\pmod{k-1}.
\end{align*}
[/guided]
[/step]
[step:Count point-pairs inside blocks to obtain the second congruence]
Define the number of blocks
\begin{align*}
b := |\mathcal{B}|,
\end{align*}
counting repeated blocks with multiplicity.
Count the set
\begin{align*}
J := \{(P,B): P\subset X, |P|=2, B\in\mathcal{B}, P\subset B\}.
\end{align*}
Choosing the pair $P$ first gives
\begin{align*}
|J|=\lambda {v\choose 2}=\lambda\frac{v(v-1)}{2}.
\end{align*}
Choosing the block $B$ first gives
\begin{align*}
|J|=b{k\choose 2}=b\frac{k(k-1)}{2}.
\end{align*}
Equating the two expressions and multiplying by $2$ yields
\begin{align*}
bk(k-1)=\lambda v(v-1).
\end{align*}
Since $b$ is an integer, $k(k-1)$ divides $\lambda v(v-1)$, equivalently
\begin{align*}
\lambda v(v-1)\equiv 0\pmod{k(k-1)}.
\end{align*}
[/step]
[step:Apply Wilson's external asymptotic existence theorem to the admissible parameters]
We use Wilson's external asymptotic existence theorem for pairwise balanced designs, in the special balanced incomplete block design form proved in R. M. Wilson, "An existence theory for pairwise balanced designs. I. Composition theorems and morphisms," Journal of Combinatorial Theory, Series A 13 (1972), 220-245; "II. The structure of PBD-closed sets and the existence conjectures," Journal of Combinatorial Theory, Series A 13 (1972), 246-273; and "III. Proof of the existence conjectures," Journal of Combinatorial Theory, Series A 18 (1975), 71-79. The special case needed here states: for every fixed pair of integers $k\ge 2$ and $\lambda\ge 1$, there exists an integer $N=N(k,\lambda)$ such that whenever $v\ge N$ and
\begin{align*}
\lambda(v-1)&\equiv 0\pmod{k-1}, & \lambda v(v-1)&\equiv 0\pmod{k(k-1)},
\end{align*}
there exists a $2-(v,k,\lambda)$ design. This cited result is the external existence input, not a consequence of the present argument. Its hypotheses in this special case are exactly the fixed integers $k\ge 2$, $\lambda\ge 1$, the lower bound $v\ge N(k,\lambda)$, and the two displayed admissibility congruences. Thus, setting
\begin{align*}
v_0(k,\lambda):=N(k,\lambda)
\end{align*}
gives the asserted sufficiency for every $v\ge v_0(k,\lambda)$.
[/step]
[step:Combine necessity and sufficiency]
The first two steps show that any $2-(v,k,\lambda)$ design forces the two displayed congruences. The Wilson existence step shows that, for every $v\ge v_0(k,\lambda)$, those same congruences force the existence of a $2-(v,k,\lambda)$ design. Therefore, for all $v\ge v_0(k,\lambda)$, existence is equivalent to the two admissibility congruences, as claimed.
[/step]
