The *Leray number* $L_{\Bbbk}(K)$ (relative to a field $\Bbbk$) of a simplicial complex $K$ is the least $d\geq 0$ such that $\widetilde H_n(C,\Bbbk)=0$ for all $n\geq d$ and all induced subcomplexes $C$ of $K$.

Leray numbers have historically arisen in at least two distinct contexts. In combinatorics they arose in the study of Helly type theorems. If $K$ can be realized as the nerve of a family of convex subsets of $\mathbb R^d$, then $L_{\Bbbk}(K)\leq d$.

They also come up in commutative algebra in the study of Stanley-Reisner rings. Namely, the Castelnuovo-Mumford regularity of the Stanley-Reisner ring of $K$ is $L_{\Bbbk}(K)$.

There are a number of basic results about Leray numbers for which I would like to know the original source. Because there are two somewhat distinct literatures on the subject it is hard to determine who first proved certain results (and perhaps some of the results were independently rediscovered). I know how the proofs of these results go, so I really just want references.

Question 1.What is the original reference for the theorem that $L_{\Bbbk}(K)\leq 1$ iff $K$ is the clique complex of a chordal graph?

Question 2.What is the original reference proving that the nerve of a family of convex sets in $\mathbb R^d$ has Leray number at most $d$? I don't believe Helly did this explicitly. Is it implicit in his paper?

Question 3.What is the original source for the connection between Leray numbers and Castelnuovo-Mumford regularity of Stanley-Reisner rings or ideals?