Let T (d, r) def = (r − 1)(d + 1) + 1 be the parameter in Tverberg’s theorem. We say that a partition I of {1, 2, . . . , T (d, r)} into r parts occurs in an ordered point sequence P if P contains a subsequence P ′ of T (d, r) points such that the partition of P ′ that is order-isomorphic to I is a Tverberg partition. We say that I is unavoidable if it occurs in every sufficiently long point se...

How many edges can be in a graph which is forced to be contained in every graph on n verti-ces and e edges? 1n this paper we obtain bounds which are in many cases asymptotically best possible. A well-known theorem of Turán [10,11] asserts that every graph on n vertices and e edges must contain a complete subgraph on m vertices if e~2(m 1)(n-rL)+(2) where r satisfies r-n (mod m-1) and In this pa...

A biologically unavoidable sequence is an infinite gender sequence which occurs in every gendered, infinite genealogical network satisfying certain tame conditions. We show that every eventually periodic sequence is biologically unavoidable (this generalizes König’s Lemma), and we exhibit some biologically avoidable sequences. Finally we give an application of unavoidable sequences to cellular ...