Online and size anti-Ramsey numbers (Q2448253)

The authors deal with the edge colorings of finite graphs and say that a coloring of a graph \(H\) is \textit{proper} if any two adjacent edges of \(H\) have distinct colors of \(H\) are of distinct colors. The \textit{sizes anti-Ramsey number} of \(H\) is defined as the smallest number \(AR_S(H)\) of edges in some graph \(G\) such that any proper coloring of \(G\) gives a rainbow copy of \(H\). Making the coloring process into a game between two players, Builder and Painter, the authors define \textit{online anti-Ramsey number} \(AR_0(H)\) and two auxiliary numbers. After a survey of related papers (in particular, on Turan numbers, Latin squares, matroids), the authors do a hard work to provide bounds on \(AR_S(H)\), \(AR_0(H)\) and auxiliary numbers for some special graphs \(H\).