2-TRANSITIVE DESIGNS

NEW

Problem: Reconstructed statement: there is no finite point-2-transitive $2$-$(v,k,\lambda)$ design with $\lambda>1$ whose points are those of an underlying finite affine/projective geometry, whose blocks are some but not all $e$-spaces of that geometry, and whose induced design-lines are exactly the geometric lines.
This matches the quoted wording: “using some but not all $e$-spaces” and “lines ... all the lines of the underlying geometry.” The wording is ambiguous about affine versus projective geometry; the affine case is a standard reading and already refutes the literal conjecture.

Result: The conjecture is false.

Let $q\ge 3$, let $V=\mathbb F_{q^4}$ viewed as a $4$-dimensional vector space over $\mathbb F_q$, and take the affine geometry $\mathrm{AG}(4,q)$ on point set $V$. Choose $\theta\in\mathbb F_{q^4}\setminus \mathbb F_{q^2}$, and put

Let

The stabilizer of $U$ in $\mathbb F_{q^4}^{*}$ is exactly $\mathbb F_q^{*}$: if $aU=U$, then $a\in U$, and multiplication by $a$ preserves $U$; if $a\notin \mathbb F_q$, then $\mathbb F_q[a]\subseteq U$, forcing $a\in\mathbb F_{q^2}$, but $U\cap\mathbb F_{q^2}=\mathbb F_q$, contradiction. Hence

Define blocks to be all affine planes

These are some but not all affine $2$-spaces: the total number of $2$-dimensional directions in $V$ is

which is larger than $|\mathcal O|=(q+1)(q^2+1)$.

Now fix two distinct points $x,y\in V$, and set $d=y-x\ne0$. Blocks containing both $x$ and $y$ are precisely the planes $x+W$ with $W\in\mathcal O$ and $d\in W$. Since $\mathbb F_{q^4}^{*}$ is transitive on $V\setminus\{0\}$, the number of such $W$ is independent of $d$. Counting incidences $(d,W)$ with $0\ne d\in W$,

Dividing by $q^4-1$ gives

Thus the construction is a $2$-$(q^4,q^2,q+1)$ design.

Moreover, the affine group

acts sharply $2$-transitively on points and preserves the block set. Finally, the intersection of all blocks containing two distinct points $x,y$ is

the ordinary affine line through $x$ and $y$. Hence the design-lines are exactly the lines of $\mathrm{AG}(4,q)$.

So a $2$-transitive design with $\lambda>1$ using some but not all $2$-spaces does exist.

Citation: No external resolution is needed; the construction above is explicit.