Peter, Jazz arrangers use the 3rd and seventh of a chord to imply the whole chord, the idea being that the sound is not ambiguous because those notes would occur together in one and only one chord--this is not exactly true, it is very clear to the ear what chord is implied-- Usually, at least in a modern melody, a single note will tell you what chord is appropriate(though there may be more than one)--The fundamental and the third give a good indication(particularly on instruments like the guitar, where the fifth is a really strong harmonic, so that it is basically there whenever the fundamental sounds, anyway--

Melodies often feature phrases that are nothing more than an arpeggio, and the chord change is indicated simply because the melody moves to a note not in the chord of the arpeggio--

Given that arrangements use chord partials(pieces of chords that imply the whole chord) all the time, it still takes three notes minimum to constitute an official chord, and the interval between those notes must be thirds of some sort or another--