Alternating cycles of length 4t+2

Determine the maximum number of alternating (4t+2)-cycles in red-blue complete graphs for every t≥1, including the alternating 6-cycle, and establish whether the random red-blue colouring is asymptotically extremal.

Background

The paper proves sharp asymptotic results for alternating cycles whose lengths are divisible by four. It observes that the corresponding bipartite extremal construction contains no alternating cycles of length 4t+2, whereas random colourings provide a nonzero density, leading the authors to leave the optimality question unresolved.