Minimum Number of Monotone Subsequences of Length 4 in Permutations

We show that for every sufficiently large n, the number of monotone subsequences of length four in a permutation on n points is at least [EQUATION PRESENTED] Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colourings of complete graphs with two colours, where the number of monochromatic K₄ is minimized. We show that all the extremal colourings must contain monochromatic K₄ only in one of the two colours. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.