Abstract
Let $f(n,r)$ denote the maximum number of colourings of $A \subseteq \lbrace 1,\ldots,n\rbrace$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $\lbrace x,y,z\rbrace$ such that $x+y=z$. We show that $f(n,2) = 2^{\lceil n/2\rceil}$, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of $f(n,r)$ for $r \leqslant 5$. Similar results were obtained by Hán and Jiménez in the setting of finite abelian groups.
Highlights
Introduction and resultsA recent trend in combinatorial number theory has been to consider versions of classical problems from extremal graph theory in the sum-free setting
Its graph-theoretic counterpart is Ramsey’s theorem from 1928 which guarantees a monochromatic clique in any r-edge-colouring of a sufficiently large complete graph
In the sum-free setting, the removal lemma of Green [15], and Kral’, Serra and Vena [27] states that every A ⊆ [n] containing o(n2) Schur triples can be made sum-free by removing o(n) elements
Summary
A recent trend in combinatorial number theory has been to consider versions of classical problems from extremal graph theory in the sum-free setting. The famous theorem of Mantel from 1907 [30] states that every n-vertex graph with more than n2/4 edges necessarily contains a triangle. The triangle removal lemma of Ruzsa and Szemeredi [33] states that every n-vertex graph containing o(n3) triangles can be made triangle-free by removing o(n2) edges. In the sum-free setting, the removal lemma of Green [15], and Kral’, Serra and Vena [27] states that every A ⊆ [n] containing o(n2) Schur triples can be made sum-free by removing o(n) elements. Kleitman and Rothschild [12] proved that the number of n-vertex triangle-free graphs is 2n2/4+o(n2), that is, the obvious lower bound of taking every subgraph of a maximal triangle-free graph is, in a sense, tight.
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