V_{j-1}$ or $x_{i-1},x_{i+1}\in V_{j-1}$ for some $j$. Clearly a path which doesn’t change direction is a dense flag.

We will use the following easy consequence of $(\Sigma 4)_{j\leq n}$.

\[l:cycle\] If $\gamma=(a,b,\ldots, b',a)$ is a simple cycle, then any two vertices $x,y\in \gamma\cap (V_{j-1}\cup V_j)$ are joined by an $E_j$-path.

Suppose that $j,j+m$ are minimal and maximal, respectively, with $\gamma\cap V_j, V_{j+m}\neq\emptyset$. Apply $(\Sigma 4)_m$ to all vertices of $\gamma\cap V_{j+m}$ and replace any path of the form $(x,y,z)$ with $y\in V_{j+m}$ by a path from $x$ to $z$ contained in $V_j\cup\ldots \cup V_{j+m-1}$. This yields a new path contained in $V_j\cup\ldots \cup V_{j+m-1}$. Next apply $(\Sigma 4)_{m-1}$ to each vertex of type $V_j$ to obtain a path in $V_{j+1}\cup\ldots \cup V_{j+m-1}$. Alternating between replacing the top and bottom extremal peaks by a path using fewer levels, we end up with an $E_k$-path for some $j\leq k\leq j+m-1$ between vertices of $\gamma$. Since an $E_k$-path changes direction at every vertex, we can apply the same procedure in order to replace this path by an $E_{k'}$-path for any $j\leq k'\leq j+m-1$.

\[def:reduced\] We call a path $\gamma=(x_0,\ldots x_k)\subseteq V_j\cup\ldots \cup V_{j+m}$ *reduced* if $\gamma$ is a flag or if the following 