Exact value of 3 color weak Rado number

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• 作者：M.P. Revuelta¹ ; ^{pastora@us.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Schur numbers ; sum-free sets ; weak Schur numbers ; weakly sum-free sets ; Rado numbers ; weak Rado numbers
• 刊名：Electronic Notes in Discrete Mathematics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：54
• 期：Complete
• 页码：241-245
• 全文大小：165 K

文摘

For integers k, n, c with k  , 1366&_mathId=si1.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=37cd048725592d0fdf634373d3d71abb" title="Click to view the MathML source">n≥1ontainer hidden">$n\ge 1$ and 1366&_mathId=si2.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=3518ef8274a5cf648f3b3e13dcdcb7c3" title="Click to view the MathML source">c≥0ontainer hidden">$c\ge 0$, the n   color weak Rado number 1366&_mathId=si3.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=e4c702ac19c53b72fa4b8a16692690be" title="Click to view the MathML source">WR_k(n,c)ontainer hidden">$W{R}_k(n,c)$ is defined as the least integer N, if it exists, such that for every n  -coloring of the set {1366&_mathId=si4.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=9005d58c9fbe0aa5e8248d8b2473a819" title="Click to view the MathML source">1,2,…,Nontainer hidden">$1,2,\dots ,N$}, there exists a monochromatic solution in that set to the equation 1366&_mathId=si5.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=5857ffae63c7a9a009f8f4921e4f1a3a" title="Click to view the MathML source">x₁+x₂+…+x_k+c=x_k+1ontainer hidden">$x_1+x_2+\dots +x_k+c=x_{k+1}$, such that 1366&_mathId=si6.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=8bda8ef0b38256d324529d7da4e473fb" title="Click to view the MathML source">x_i≠x_jontainer hidden">$x_i\ne x_j$ when 1366&_mathId=si7.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=8d9851978ddca93589259301baee5a70" title="Click to view the MathML source">i≠jontainer hidden">$i\ne j$. If no such N   exists, then 1366&_mathId=si3.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=e4c702ac19c53b72fa4b8a16692690be" title="Click to view the MathML source">WR_k(n,c)ontainer hidden">$W{R}_k(n,c)$ is defined as infinite.

In this work, we consider the main issue regarding the 3 color weak Rado number for the equation 1366&_mathId=si8.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=d4a200a27f49531d913847e47fd480c4" title="Click to view the MathML source">x₁+x₂+c=x₃ontainer hidden">$x_1+x_2+c=x_3$ and the exact value of the 1366&_mathId=si9.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=41dc7a713bbcf6ec15863c28aa1431aa" title="Click to view the MathML source">WR₂(3,c)=13c+22ontainer hidden">$W{R}_2(3,c)=13c+22$ is established.