Flutters and Chameleons
Wednesday, 30.10.19, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Let K be the collection of infinite sets of natural numbers. A colouring c\nof K with a finite number of colours is Ramsey if for some infinite A in K and every infinite subset B of A, c(B) = c(A).\nA non-Ramsey colouring is one for which no such A exists.\nSolovay in a famous paper published in 1970 used a strongly inaccessible cardinal to construct a model of ZF + DC in which various principles hold which contradict AC:\n\n\nLM: every set of real numbers is Lebesgue measurable;\n\nPB: every set of real numbers has the property of Baire;\nUP: every uncountable set of real numbers has a perfect subset.\nTwo other principles to be considered are\n\nRAM: all colourings are Ramsey\nNoMAD: there is no maximal infinite family of pairwise almost disjoint infinite sets of natural numbers.\n\n\nThe speaker showed in 1968 that in Solovay's model, RAM holds, and in 1969 that if one started from a Mahlo cardinal, NoMAD would hold in the corresponding Solovay model.\nIt is natural to ask whether these large cardinals are necessary; the inaccessible is necessary for UP (Specker) and LM (Shelah) but not for PB (Shelah).\nMore recently Toernquist has shown that NoMAD holds in Solovay's original model, and Shelah and Horowitz have extended his work to show that even that inaccessible is unnecessary to get a model of NoMAD. Toernquist and Schrittesser have very recently shown that NoMAD follows from RAM plus a uniformisation principle.\n\nBut it has been open for fifty years whether RAM requires an inaccessible.\nThis talk will be chiefly about flutters and chameleons, which are non-Ramsey sets with elegant properties, constructed using weak forms of AC; surprisingly their existence has been found to follow from various Pareto principles of mathematical economics, as described in this week's colloquium talk by Giorgio Laguzzi.\n\nTheir relation to feeble filters will also be discussed: that every free filter on the set of natural numbers is feeble follows from RAM (Mathias 1973) but not from NoMAD, (Shelah and Horowitz in a second recent paper). A filter is feeble if it is meagre in the Cantor topology; equivalently, if some finjection projects it to the Frechet filter.\n\n\n