Decomposing complete graphs into clique-free graphs of certain size Modified in accordance with Tsuyoshi's comment which seems to generalize.

Let $K_{m}$ be a complete graph on $m$ vertices. Is there a way to partition the graphs in to sets of graphs that have no cliques of size $k$ for some $k \in \mathbb{Z}_{+}$? What is the minimum number of partitions would I need to make?

How about the case when $K_{n}$ is replaced by a bipartite $K_{m,n}$ where one seeks a partition such that there is no $K_{k,l}$ for some $k,l \in \mathbb{Z}_{+}$? What is the minimum number of partitions would I need to make?
This question is about Ramsey theory.  In the case of the complete graphs, you can take a=1 by considering partitioning into only two sets.

The Ramsey number R(k,k) is the minimum integer m such that however you partition the edges of the complete graph Km into two sets, at least one of the sets contains Kk as a subgraph.  This immediately implies that if m &lt; R(k,k), then the edges of the complete graph Km can be partitioned into two sets so that neither set contains Kk as a subgraph.

Erdős proved that R(k,k) grows exponentially: R(k,k) &gt; k⋅2k/2/(e√2).  This means that if m is sufficiently large, then m &lt; R(k,k) for k = ⌈2 log2m⌉ = O(log m).  In other words, you can take a=1 with two sets.