Subgraphs of a hypercube containing no small even cycles

We investigate several Ramsey‐Turán type problems for subgraphs of a hypercube. We obtain upper and lower bounds for the maximum number of edges in a subgraph of a hypercube containing no four‐cycles or more generally, no 2 k ‐cycles C 2k . These extermal results imply, for example, the following Ramsey theorems for hypercubes: A hypercube can always be edge‐partitioned into four subgraphs, each of which contains no six‐cycle. However, for any integer t , if the n ‐cube is edge‐partitioned into t sub‐graphs, then one of the subgraphs must contain an edight‐cycle, provided only than n is sufficiently large (depending only on t ).