Sharp threshold for fractional triangle decompositions in random graphs

Determine whether, for every fixed positive constant and edge probability at least a factor of 1 plus that constant times the sharp threshold for every edge to lie in a triangle, the binomial random graph admits a fractional triangle decomposition with high probability.

Background

The paper studies fractional clique decompositions of sparse random graphs and hypergraphs. It proves that a binomial random graph G(n,p) has a fractional triangle decomposition with high probability when p is at least n{-1/2+o(1)}, which is optimal up to a subpolynomial factor relative to the necessary condition that every edge lie in a triangle.

The conjectured sharp threshold is smaller by a logarithmic factor. Thus, although the paper establishes the correct polynomial exponent, it does not resolve the exact threshold conjectured by Mahabaduge and Simkin.

References

It is widely believed that this condition should also be sufficient, and Mahabaduge and Simkin explicitly conjectured the following sharp threshold in the triangle case, in response to the problem of Yuster.

\begin{conjecture}[Mahabaduge and Simkin] \label{conj:triangle_threshold} For every $ > 0 $ and $ p \ge (1 + ) \sqrt{\frac{3 \log{n}{2n} $, w.h.p.\ $ G(n, p) $ admits a fractional triangle decomposition. \end{conjecture}

— Fractional clique decompositions in random hypergraphs  (2609.29943 - Joos et al., 24 Sep 2026) in Section 1, Introduction; Conjecture attributed to Mahabaduge and Simkin, labeled Conjecture 1

We believe that the analogue of \cref{conj:triangle_threshold} holds in general for fractional clique decompositions in random hypergraphs.

Write $$ p*_{k, r}(n) \coloneqq c_{k, r} \left( \frac{\log{n}{n{r - k} \right){\frac{1}{\binom{r}{k} - 1} \quad \text{where} \quad c_{k, r} \coloneqq \left( \left(k-\frac{r-k}{\binom{r}{k}-1}\right)(r-k)! \right){\frac{1}{\binom{r}{k}-1} $ \quad \text{is the unique constant such that } p*_{k, r} \text{ is the sharp threshold function for the property that every edge present in } G{(k)}(n, p) \text{ is contained in a copy of } K_r{(k)}.

\begin{conjecture} \label{conj:general_threshold} Let $ k \ge 2 $, $ r \ge k + 1 $, and $ > 0 $, and suppose $ p \ge (1 + ) p*_{k, r}(n) $. Then w.h.p.\ $ G{(k)}(n, p) $ admits a fractional $ K_r{(k)} $-decomposition. \end{conjecture}

— Fractional clique decompositions in random hypergraphs  (2609.29943 - Joos et al., 24 Sep 2026) in Section 6, Concluding remarks; Conjecture labeled Conjecture 2

We wonder whether a suitable variant of the process defined in this paper would also converge to a fractional clique decomposition at (or a constant factor above) the conjectured threshold.

— Fractional clique decompositions in random hypergraphs  (2609.29943 - Joos et al., 24 Sep 2026) in Section 6, Concluding remarks