99 research outputs found
A nilpotent IP polynomial multiple recurrence theorem
We generalize the IP-polynomial Szemer\'edi theorem due to Bergelson and
McCutcheon and the nilpotent Szemer\'edi theorem due to Leibman. Important
tools in our proof include a generalization of Leibman's result that polynomial
mappings into a nilpotent group form a group and a multiparameter version of
the nilpotent Hales-Jewett theorem due to Bergelson and Leibman.Comment: v4: switch to TeXlive 2016 and biblate
Forbidding intersection patterns between layers of the cube
A family is said to be an antichain
if for all distinct . A classic result
of Sperner shows that such families satisfy , which is easily seen to be best possible. One can
view the antichain condition as a restriction on the intersection sizes between
sets in different layers of . More generally one can ask,
given a collection of intersection restrictions between the layers, how large
can families respecting these restrictions be? Answering a question of Kalai,
we show that for most collections of such restrictions, layered families are
asymptotically largest. This extends results of Leader and the author.Comment: 16 page
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