Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, May 27, 2021, 12:15 pm

Duration: 30 minutes

Location: Zoom: conference room

Speaker: Charlotte Knierim

Schur's Theorem for randomly perturbed sets

A set of integers A is said to be Schur if any two-colouring of A results in monochromatic x, y and z with x+y=z. We study the following problem: How many random integers from [n] need to be added to some A⊆[n] to ensure that the resulting set is Schur with high probability? Hu showed in 1980 that when |A| > ⌈4n/5⌉, no random integers are needed as A is already guaranteed to be Schur. Recently, Aigner-Horev and Person showed that for any dense set of integers A⊆[n], adding ω(n^1/3) random integers suffices, noting that this is optimal for sets A with |A|≤⌈n/2⌉. Here we complete the picture by closing the gap between these two results. We show that if A⊆[n], with |A|=⌈n/2⌉+t > ⌈4n/5⌉ then adding ω(min{n^1/3,nt^-1}) random integers will result in a set that is Schur with high probability. We also obtain some results on the sparse transition.

This is joint work with Shagnik Das and Patrick Morris from FU Berlin.

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