Let $\gamma: [a,b]\to\mathbb{R}^d$ be a smooth (i.e. $\gamma\in C^\infty (\mathbb{R}))$ and regular ($\gamma^\prime(t)\neq \vec 0$) curve with arc-length defined by $$\gamma(t)=(\gamma_1(t),\dots,\gamma_d(t)).$$

Assume that $\gamma_j(t),\gamma_j^\prime(t)\neq 0$ for $1\le j\le d$.

Define a partition $P=\{a,x_0,x_1,\dots,b\}$ on $[a,b]$ by: $$ x_m= \max_{t\in[x_{m-1},b]} \{\forall s\in [a,x_{m-1}],\gamma_j(t)\neq\gamma_j(s) \quad \forall 1\le j\le d-1\}. $$ Roughly speaking, this partition divides $\gamma(t)$ to injective graphs in respect to last coordinate.

My question is "how large can P be?". To put it in other words, Can we find a curve $\gamma$ with $|P| \ge \omega$? ${\mathfrak c}$? ($|P|$ denotes the ordinal of the set). Can we bound this ordinal?