For a positive integer n, let [n] denote {1, …, n}. For a 2-dimensional integer lattice point b and positive integers k ≥ 2 and n, a k-sum b-free set of [n] × [n] is a subset S of [n] × [n] such that there are no elements a1, …, ak in S satisfying a1 + · · · + ak = b. For a 2-dimensional integer lattice point b and positive integers k ≥ 2 and n, we determine the maximum density of a k-sum b-free set of [n] × [n]. This is the first investigation of the non-homogeneous sum-free set problem in higher dimensions.
Supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2018R1C1B6003577). Supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07043049), and also by Hankuk University of Foreign Studies Research Fund. The authors thank Hong Liu at the University of Warwick for drawing our attention to this area. This work was done during the 3rd Korean Early Career Researcher Workshop in Combinatorics.∗Supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07043049), and also by Hankuk University of Foreign Studies Research Fund. †Corresponding author. Supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)(NRF-2018R1C1B6003786). ‡Supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2018R1C1B6003577).
