Discrete Geometry and Extremal Graph Theory - IMSA

We prove the following result using a similar technique.
Theorem 2. A graph G on n vertices without a “Θ-graph” contains at most c ′ n 3/2 edges
for some constant c ′ > 0.
Proof of Theorem 2 (sketch). As in the proof of Theorem 1, let d 1 , d 2 , . . . , d n be the degrees
of the vertices v 1 , v 2 , . . . , v n . For a vertex v i , we count the pairs of the d i vertices
connected to v i . As we continue this count for all the vertices of G, note that each pair is
counted at most twice, or else G must have a “Θ-graph.” In other words, we arrive at the
inequality
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2)
2
+
2
+ · · ·
Thus, as in the proof of Theorem 1, we see c ′ ≈ √ 2c will work for sufficiently large n.
Finally, we prove our main result, showing an unexpected connection between Erdős’ problem
on unit distances and graph theoretic results.
Theorem 3. Let f(n) be the maximum number of unit distances among n points in the
plane. Then, f(n) ≤ c ′ n 3/2 .
Proof of Theorem 3. Consider the unit distance graph of the n points defined as follows: two
points are joined by a segment if and only if they are unit distance apart. Now, note that
no “Θ-graph” can appear in such a graph, for let us consider the two end vertices. The two
unit circles centered at these vertices must each pass through three common points, but two
circles can intersect in at most two points, contradiction. Thus, by Theorem 2, the result
follows.
2
≤ 2
.
3