2022/05/03 by Das, Shagnik, Knierim, Charlotte, Morris, Patrick · 1 citation
#05D15 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2205.01456
A set A of integers 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 with high probability that the resulting set is Schur? Hu showed in 1980 that when |A|> \lceil\tfrac4n5\rceil, 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(n1/3) random integers suffices, noting that this is optimal for sets A with |A|≤ \lceil\tfracn2\rceil. We close the gap between these two results by showing that if A⊆ [n] with |A|=\lceil\tfracn2\rceil+t