deg ree ▁, ▁and ▁in ▁which ▁the ▁edges ▁are ▁- color ed ▁in ▁such ▁a ▁way ▁that ▁every ▁two ▁edges ▁with ▁the ▁same ▁source ▁vertex ▁have ▁distinct ▁colors . ▁The ▁road ▁color ing ▁problem ▁is ▁the ▁problem ▁of ▁edge - color ing ▁a ▁directed ▁graph ▁with ▁uniform ▁out - de gre es , ▁in ▁such ▁a ▁way ▁that ▁the ▁resulting ▁automat on ▁has ▁a ▁synchron izing ▁word . ▁ ▁solved ▁the ▁road ▁color ing ▁problem ▁by ▁prov ing ▁that ▁such ▁a ▁color ing ▁can ▁be ▁found ▁whenever ▁the ▁given ▁graph ▁is ▁strongly ▁connected ▁and ▁a period ic . ▁ ▁Ram sey ' s ▁theorem ▁concerns ▁the ▁problem ▁of ▁- color ing ▁the ▁edges ▁of ▁a ▁large ▁complete ▁graph ▁ ▁in ▁order ▁to ▁avoid ▁creating ▁mon och rom atic ▁complete ▁sub graph s ▁ ▁of ▁some ▁given ▁size   . ▁According ▁to ▁the ▁theorem , ▁there ▁exists ▁a ▁number ▁ ▁such ▁that , ▁whenever ▁, ▁such ▁a ▁color ing ▁is ▁not ▁possible . ▁For ▁instance , ▁, ▁that ▁is , ▁if ▁the ▁edges ▁of ▁the ▁graph ▁ ▁are ▁ 2 - color ed , ▁there ▁will ▁always ▁be ▁a ▁mon och rom atic ▁triangle . ▁ ▁A ▁path ▁in ▁an ▁edge - color ed ▁graph ▁is ▁said ▁to ▁be ▁a ▁rain bow ▁path ▁if ▁no ▁color ▁repe ats ▁on ▁it . ▁A ▁graph ▁is ▁said ▁to ▁be ▁rain bow ▁colored ▁if ▁there ▁is ▁a ▁rain bow ▁path ▁between ▁any ▁two ▁pairs ▁of ▁vertices . ▁An ▁edge - col our ing ▁of ▁a ▁graph ▁G