Bradač–Liu–Wu–Xu conjecture on admissible colorings of ordered cliques

Let $G$ be an ordered clique whose vertices are ordered as $v_1<\cdots<v_N$, and let $\chi$ be a red-blue edge-coloring of $G$. Its dependency digraph $D(\chi)$ has vertex set $\{v_1,\ldots,v_N\}$ and, for every $i\in[N-1]$ and $j\in[N]\setminus\{i,i+1\}$, contains the directed edges $(v_j,v_i)$ and $(v_j,v_{i+1})$ whenever $\chi(v_iv_{i+1})$ is red and $\chi(v_iv_j)=\chi(v_{i+1}v_j)$ is blue. The coloring is admissible if $D(\chi)$ is acyclic. For $k\geq 1$, let $f(k)$ be the minimum integer $N$ such that every red-blue edge-coloring of the ordered clique on $N$ vertices contains $k$ vertices inducing an admissible coloring.

Bradač–Liu–Wu–Xu conjecture. For every integer $k\geq 1$,

$$f(k)=\left\lfloor\frac{k^2}{4}\right\rfloor+1.$$

This conjecture gives the expected sharp threshold for finding admissible induced ordered subcliques in two-colored ordered cliques. The source attributes the conjecture to Bradač et al.; no resolution status is supplied here.