Saturated $k$-Sperner Systems of Minimum Size ★★

Author(s): Morrison; Noel; Scott

Question   Does there exist a constant and a function such that if , then every saturated -Sperner system has cardinality at least ?

List Colourings of Complete Multipartite Graphs with 2 Big Parts ★★

Author(s): Allagan

Question   Given , what is the smallest integer such that ?

Geometric Hales-Jewett Theorem ★★

Author(s): Por; Wood

Conjecture   For all integers and , there is an integer such that for every set of at least points in the plane, if each point in is assigned one of colours, then:
\item contains collinear points, or \item contains a monochromatic line (that is, a maximal set of collinear points receiving the same colour)

Keywords: Hales-Jewett Theorem; ramsey theory

Generalised Empty Hexagon Conjecture ★★

Author(s): Wood

Conjecture   For each there is an integer such that every set of at least points in the plane contains collinear points or an empty hexagon.

Keywords: empty hexagon