What is the purpose of the note on the chalkboard?

From the title and throughout the story, culminating in the scrawled threat on the blackboard, there are layers of irony that merit analysis. "La hôte" is the French title of the story, meaning either "the guest" or "the host." The fact that Daru refers to the prisoner alternately as a "guest" and a "prisoner" leads us to question how he really regards the Arab. Daru learns from Balducci that the Arab killed his cousin in "a family squabble." Daru feels then "a sudden wrath . . . against all men with their rotten spite, their tireless hates, their blood lusts."
What of the fact that the note on the blackboard is drawn where Daru has also drawn "the four rivers of France in four different coloured chalks"? Why is he not teaching his impoverished students the geography of their own colonized country? Daru admits that "everywhere else" but Algeria (where he was born), "he felt exiled." But perhaps he is also exiled here: "No one in this desert, neither he nor his guest, mattered." And thinking of Balducci, "he could still hear the gendarme's farewell and without knowing why, he felt strangely empty and vulnerable." And of course he is both empty and vulnerable, and the purpose of the note is to suggest that both "brotherhood" and Daru's giving the Arab freedom to choose his fate are false notions. The last lines of the story resonate with his ultimate sense of alienation and exile:

Daru looked at the sky, the plateau, and beyond the invisible lands stretching all the way to the sea. In this vast landscape he had loved so much, he was alone.

The note on the chalkboard has a number of purposes. First, the note brings the story to an end. After releasing the Arab and enabling him to make a free choice, Daru returns to the schoolhouse to find this threatening note from the Arab's allies, thus bringing the story to its conclusion.
Looking deeper, however, we see that this note has a thematic purpose because it links to the theme of alienation, which is expressed through the character of Daru. Daru is not just alienated from society because of his physical location in the mountains, he is also alienated because he does not want to take sides between the French colonial government and the Arabs. This note, therefore, reinforces his alienation: it shows that no matter how hard he tries to avoid this conflict, he cannot help but be drawn into it. He is pushed further and further from both the Arabs and the French, thereby becoming an outsider.
In addition, the note also links to the theme of free will and choice. While Daru believed that allowing the prisoner to make his own choice was the right thing to do, this note proves otherwise. Camus, therefore, uses the note to question the concept of free choice for people who live under colonial regimes. Every choice has a political consequence, suggesting that free will might be little more than a myth.