$KG(\mathfrak a_1,\mathfrak b_1)\cong KG(\mathfrak a_2,\mathfrak b_2)$ with $(\mathfrak a_1,\mathfrak b_1)\neq(\mathfrak a_2,\mathfrak b_2)$

Question

Let $\mathfrak a,\mathfrak b$ be cardinals (a cardinal being identified with the smallest ordinal of that cardinality), $\mathfrak b\subseteq\mathfrak a$. The Kneser graph $KG(\mathfrak a,\mathfrak b)$ is the simple graph with vertex set $V(KG(\mathfrak a,\mathfrak b))=\binom{\mathfrak a}{\mathfrak b}:=\{X\subseteq \mathfrak{a}:|X|=\mathfrak b\}$ and $XY\in E(KG(\mathfrak a,\mathfrak b))$ iff $X\cap Y=\emptyset$.

Do there exist cardinal pairs $(\mathfrak a_1,\mathfrak b_1)\neq(\mathfrak a_2,\mathfrak b_2)$ with $\mathfrak a_1,\mathfrak a_2$ infinite and $\mathfrak b_1,\mathfrak b_2\neq 0$ such that $KG(\mathfrak a_1,\mathfrak b_1)$ and $KG(\mathfrak a_2,\mathfrak b_2)$ are isomorphic, in short $KG(\mathfrak a_1,\mathfrak b_1)\cong KG(\mathfrak a_2,\mathfrak b_2)$?

Answer 1

No. If $G=KG(\mathfrak a,\mathfrak b)$, where $\mathfrak a$ is infinite and $0\lt\mathfrak b\le\mathfrak a$, then $\mathfrak a$ and $\mathfrak b$ are determined by the structure of the graph $G$.

$\mathfrak a$ is the maximum cardinality of a clique (set of pairwise adjacent vertices) in $G$.

$\mathfrak b=1$ if and only if $G$ is a complete graph. If $\mathfrak b\gt1$, then $\mathfrak b$ is the maximum cardinality of a clique in $G$ for which there is a vertex of $G$ which is adjacent to no element of the clique.