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For integers m>km>, m>nm>, m>cm> with m>k m>, pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1571065316301366&_mathId=si1.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=37cd048725592d0fdf634373d3d71abb" title="Click to view the MathML source">n≥1pan>pan class="mathContainer hidden">pan class="mathCode"><math altimg="si1.gif" overflow="scroll"><mi>nmi><mo>≥mo><mn>1mn>math>pan>pan>pan> and pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1571065316301366&_mathId=si2.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=3518ef8274a5cf648f3b3e13dcdcb7c3" title="Click to view the MathML source">c≥0pan>pan class="mathContainer hidden">pan class="mathCode"><math altimg="si2.gif" overflow="scroll"><mi>cmi><mo>≥mo><mn>0mn>math>pan>pan>pan>, the m>n m> color weak Rado number pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1571065316301366&_mathId=si3.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=e4c702ac19c53b72fa4b8a16692690be" title="Click to view the MathML source">WRk(n,c)pan>pan class="mathContainer hidden">pan class="mathCode"><math altimg="si3.gif" overflow="scroll"><mi>Wmi><msub><mrow><mi>Rmi>mrow><mrow><mi>kmi>mrow>msub><mo stretchy="false">(mo><mi>nmi><mo>,mo><mi>cmi><mo stretchy="false">)mo>math>pan>pan>pan> is defined as the least integer m>Nm>, if it exists, such that for every m>n m>-coloring of the set {pan id="mmlsi4" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1571065316301366&_mathId=si4.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=9005d58c9fbe0aa5e8248d8b2473a819" title="Click to view the MathML source">1,2,…,Npan>pan class="mathContainer hidden">pan class="mathCode"><math altimg="si4.gif" overflow="scroll"><mn>1mn><mo>,mo><mn>2mn><mo>,mo><mo>…mo><mo>,mo><mi>Nmi>math>pan>pan>pan>}, there exists a monochromatic solution in that set to the equation pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1571065316301366&_mathId=si5.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=5857ffae63c7a9a009f8f4921e4f1a3a" title="Click to view the MathML source">x1+x2+…+xk+c=xk+1pan>pan class="mathContainer hidden">pan class="mathCode"><math altimg="si5.gif" overflow="scroll"><msub><mrow><mi>xmi>mrow><mrow><mn>1mn>mrow>msub><mo>+mo><msub><mrow><mi>xmi>mrow><mrow><mn>2mn>mrow>msub><mo>+mo><mo>…mo><mo>+mo><msub><mrow><mi>xmi>mrow><mrow><mi>kmi>mrow>msub><mo>+mo><mi>cmi><mo>=mo><msub><mrow><mi>xmi>mrow><mrow><mi>kmi><mo>+mo><mn>1mn>mrow>msub>math>pan>pan>pan>, such that pan id="mmlsi6" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1571065316301366&_mathId=si6.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=8bda8ef0b38256d324529d7da4e473fb" title="Click to view the MathML source">xi≠xjpan>pan class="mathContainer hidden">pan class="mathCode"><math altimg="si6.gif" overflow="scroll"><msub><mrow><mi>xmi>mrow><mrow><mi>imi>mrow>msub><mo>≠mo><msub><mrow><mi>xmi>mrow><mrow><mi>jmi>mrow>msub>math>pan>pan>pan> when pan id="mmlsi7" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1571065316301366&_mathId=si7.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=8d9851978ddca93589259301baee5a70" title="Click to view the MathML source">i≠jpan>pan class="mathContainer hidden">pan class="mathCode"><math altimg="si7.gif" overflow="scroll"><mi>imi><mo>≠mo><mi>jmi>math>pan>pan>pan>. If no such m>N m> exists, then pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1571065316301366&_mathId=si3.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=e4c702ac19c53b72fa4b8a16692690be" title="Click to view the MathML source">WRk(n,c)pan>pan class="mathContainer hidden">pan class="mathCode"><math altimg="si3.gif" overflow="scroll"><mi>Wmi><msub><mrow><mi>Rmi>mrow><mrow><mi>kmi>mrow>msub><mo stretchy="false">(mo><mi>nmi><mo>,mo><mi>cmi><mo stretchy="false">)mo>math>pan>pan>pan> is defined as infinite.p><p id="sp0020">In this work, we consider the main issue regarding the 3 color weak Rado number for the equation pan id="mmlsi8" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1571065316301366&_mathId=si8.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=d4a200a27f49531d913847e47fd480c4" title="Click to view the MathML source">x1+x2+c=x3pan>pan class="mathContainer hidden">pan class="mathCode"><math altimg="si8.gif" overflow="scroll"><msub><mrow><mi>xmi>mrow><mrow><mn>1mn>mrow>msub><mo>+mo><msub><mrow><mi>xmi>mrow><mrow><mn>2mn>mrow>msub><mo>+mo><mi>cmi><mo>=mo><msub><mrow><mi>xmi>mrow><mrow><mn>3mn>mrow>msub>math>pan>pan>pan> and the exact value of the pan id="mmlsi9" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1571065316301366&_mathId=si9.gif&_user=111111111&_pii=S1571065316301366&_rdoc=1&_issn=15710653&md5=41dc7a713bbcf6ec15863c28aa1431aa" title="Click to view the MathML source">WR2(3,c)=13c+22pan>pan class="mathContainer hidden">pan class="mathCode"><math altimg="si9.gif" overflow="scroll"><mi>Wmi><msub><mrow><mi>Rmi>mrow><mrow><mn>2mn>mrow>msub><mo stretchy="false">(mo><mn>3mn><mo>,mo><mi>cmi><mo stretchy="false">)mo><mo>=mo><mn>13mn><mi>cmi><mo>+mo><mn>22mn>math>pan>pan>pan> is established.
