Weak and strong conjectures for acyclic 0–1 patterns

Let $\mathcal{P}_{\text{acyclic}}$ be the class of acyclic 0–1 patterns. Acyclic-pattern conjecture. Both of the following forms are proposed:

• Weak Form: For all $P\in\mathcal{P}_{\text{acyclic}}$,

$$\operatorname{Ex}(P,n)=n^{1+o(1)}.$$

• Strong Form: For all $P\in\mathcal{P}_{\text{acyclic}}$, there exists a constant $C_P$ such that

$$\operatorname{Ex}(P,n)=O\bigl(n2^{C_P\sqrt{\log n}}\bigr).$$

The weak form asserts near-linearity, while the strong form gives a specific subpolynomial factor. The paper calls the strong form more plausible in light of the known edge-ordered bound, but leaves both forms as conjectural.