Log24

The 15  2-subsets of a 6-set correspond to the 15 points of PG(3,2).
(Cullinane, 1986*)

The 35  3-subsets of a 7-set correspond to the 35 lines of PG(3,2).
(Conwell, 1910)

The 56  3-subsets of an 8-set correspond to the 56 spreads of PG(3,2).
(Seidel, 1970)

Each correspondence above may have been investigated earlier than
indicated by the above dates , which are the earliest I know of.

See also Correspondences in this journal.

* The above 1986 construction of PG(3,2) from a 6-set also appeared
in the work of other authors in 1994 and 2002 . . .

• Gonzalez-Dorrego, Maria R. (Maria del Rosario),
  (16,6) Configurations and Geometry of Kummer Surfaces in  P³.
  American Mathematical Society, Providence, RI, 1994.
• Dolgachev, Igor, and Keum, JongHae,
  "Birational Automorphisms of Quartic Hessian Surfaces."
  Trans. Amer. Math. Soc. 354 (2002), 3031-3057.

Addendum at 5:09 PM suggested by an obituary today for Stephen Joyce:

See as well the word correspondences  in
"James Joyce and the Hermetic Tradition," by William York Tindall
(Journal of the History of Ideas , Jan. 1954).