اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn the maximum number of integer colourings with forbidden monochromatic sums
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Let $f(n,r)$ denote the maximum number of colourings of $A subseteq lbrace 1,ldots,nrbrace$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $lbrace x,y,zrbrace$ such that $x+y=z$. We show that $f(n,2) = 2^{lceil n/2rceil}$, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of $f(n,r)$ for $r leq 5$. Similar results were obtained by H`an and Jimenez in the setting of finite abelian groups.
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Let $mathbf{k} := (k_1,dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,dots,s$ such that, for every $c in {1,dots,s}$, the edges of colour $c$ cont
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