Permanent problem IDPPL 046
Combinatorics
Let α∈[0,1/2) and n,t≥ 1. Let F^((t))(n,α) be the largest m such that we can 2-colour the edges of the complete t-uniform hypergraph on n vertices such that if X⊆ [n] with | X| ≥ m then there are at least α C(| X|, t) many t-subsets of X of each colour. For fixed n,t as we change α from 0 to 1/2 does F^((t))(n,α) increase continuously or are there jumps? Only one jump?
Let α∈[0,1/2) and n,t≥ 1. Let F^((t))(n,α) be the largest m such that we can 2-colour the edges of the complete t-uniform hypergraph on n vertices such that if X⊆ [n] with | X| ≥ m then there are at least α C(| X|, t) many t-subsets of X of each colour. For fixed n,t as we change α from 0 to 1/2 does F^((t))(n,α) increase continuously or are there jumps? Only one jump?
A solution must appear in a reputable journal, with documentation that Erdős offered the displayed amount. Claims are administered by the Combinatorics Foundation; erdosproblems.com does not pay awards.