Zoom link: https://huji.zoom.us/j/89497187541?pwd=rwJjvOFpY2KoGIYC8K4D3FRQ5pPbYJ.1
Title: Kay-graphs and k-Ramsey
Abstract: The property for structures and classes of finite structures is closely connected to classification theory and the dividing lines in model theory. Indeed, many dividing lines, such as stability and dependence, are characterised by the behaviour of generalised indiscernibles indexed by these structures.
In our pursuit to identify dividing lines that extend beyond binary categorizations—specifically non-ternary frameworks like NIP_2 Ramsey structures—we introduce a new concept: k-Ramsey or Ramsey up-to k. This notion represents a weakening of the Ramsey property, as it is confined to colourings of structures of size up to k.
We present a family of reducts of k-hypergraphs, which we refer to as kay-graphs. This family demonstrates that k-Ramsey does not imply (k+1)-Ramsey, thereby highlighting a nuanced hierarchy within Ramsey properties.
Following this, we will discuss the implications of these findings concerning the behaviour of generalized indiscernibles indexed by structures possessing such properties.
All properties and notions will be clearly defined; no prior knowledge of Ramsey theory or generalised indiscernibles will be assumed.