论文标题
添加剂集的Schur程度
The Schur degree of additive sets
论文作者
论文摘要
让（G， +）成为一个Abelian群体。如果g的子集不包含X，y，z的子集，则x +y = z。我们通过引入g的Schur度量来扩展此概念，其中Schur度量1对应于Sumfree。经典的不平等s（n）$ \ le $ r n（3）-2，在Schur编号S（n）和Ramsey Number r n（3）= r（3，。3，3，3）之间，在更广泛的环境中保持有效，涉及G.递归上限的某些子集的Schur程度。我们制定了一个猜想，如果为，则可以填补这一空白。的确，我们对Schur学位的研究使我们猜想了所有n $ \ ge $ 2的$ \ le $ n（s（n -1） + 1）$ 2。 s（6）$ \ le $ 966猜想，而到目前为止，一切都为536 $ \ le $ s（6）$ \ le $ 1836。
Let (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x +y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The classical inequality S(n) $\le$ R n (3) -- 2, between the Schur number S(n) and the Ramsey number R n (3) = R(3,. .. , 3), is shown to remain valid in a wider context, involving the Schur degree of certain subsets of G. Recursive upper bounds are known for R n (3) but not for S(n) so far. We formulate a conjecture which, if true, would fill this gap. Indeed, our study of the Schur degree leads us to conjecture S(n) $\le$ n(S(n -- 1) + 1) for all n $\ge$ 2. If true, it would yield substantially better upper bounds on the Schur numbers, e.g. S(6) $\le$ 966 conjecturally, whereas all is known so far is 536 $\le$ S(6) $\le$ 1836.
