Is there any kind of partition like the Gallai partition of rainbow triangle - free colorings? It would necessarily be a much weaker structure.
Can we get an upper bound on the number of colors that can induce a connected subgraph? This bound would be at least 4.
Look at the following paper: http://www.combinatorics.org/ojs/index.php/eljc/article/viewFile/v19i4p33/pdf in theorem 4.
Thanks Diamantis but I'm hoping to get some information about forbidding a rainbow C_4, this paper is about no properly colored C_4. Actually the quest for information about no rainbow C_4 was part of the motivation for that paper. :-) I believe it's a hard problem.
N. Alon, On a onje ture of Erdos, Simonovits, and Sos on erning anti-Ramsey theorems, J.
Graph Theory 1 (1983), 91-94.
Jiang, Tao; West, Douglas B. On the Erdős-Simonovits-Sós conjecture about the anti-Ramsey number of a cycle. Special issue on Ramsey theory. Combin. Probab. Comput. 12 (2003), no. 5-6, 585–598.
the above papers related to ainbow C_k free graphs
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