Daisies and other Turán problems
Daisy, a hypergraph and other Turán problems in the hypercube are studied. A daisy, or r-daisy, is a certain r-uniform hypergraph consisting of six sets. Given an (r - 2)-set P and a 4-set Q disjoint from P, the daisy on (P,Q) consists of the r-sets A with P ⊂ A ⊂ P ∪ Q. A beautiful conjecture of Johnson and Talbot is proposed, about meeting d-cubes in several points, that is also closely tied to daisy problems. They conjecture that if there is a positive fraction of the vertices of the n-cube then (for n sufficiently large) there must be some d-cube containing the least points of the family.
Combinatorics Probability and Computing
Bollobás, B., Leader, I., & Malvenuto, C. (2011). Daisies and other Turán problems. Combinatorics Probability and Computing, 20 (5), 743-747. https://doi.org/10.1017/S0963548311000319