Leila Sloman writes about five mathematicians at ETH Zurich who, while pursuing a different problem in late 2025, stumbled upon a simple proof of the supercritical sharpness conjecture for all infinite transitive graphs—a decades-old question in percolation theory about how quickly networks flood past a critical threshold. The proof confirms that above the critical probability, fluid covers nearly the entire graph, resolving what researchers had called "the one remaining fortress" in the field.
- Percolation theory originated from Rosalind Franklin's 1940s work on coal porosity at the British Coal Utilization Research Association
- Oded Schramm, who co-initiated the study of percolation on transitive graphs with Itai Benjamini, died in a hiking fall in 2008 at age 46
- The key insight was reordering a standard "sprinkling" technique—analyzing the sprinkled edges first rather than last—which both simplified the proof and made it general enough for all transitive graphs
- The subcritical half of the sharpness conjecture had already been proved in 2007 by Antunović and Veselić
- A major open question remains: what happens exactly at the critical probability on three-dimensional lattices
- Percolation theory originated from Rosalind Franklin's 1940s work on coal porosity at the British Coal Utilization Research Association
- Oded Schramm, who co-initiated the study of percolation on transitive graphs with Itai Benjamini, died in a hiking fall in 2008 at age 46
- The key insight was reordering a standard "sprinkling" technique—analyzing the sprinkled edges first rather than last—which both simplified the proof and made it general enough for all transitive graphs
- The subcritical half of the sharpness conjecture had already been proved in 2007 by Antunović and Veselić
- A major open question remains: what happens exactly at the critical probability on three-dimensional lattices
Mathematicians have made significant progress on the long-stalled problem of improving bounds for near-diagonal Ramsey numbers by upgrading Paul Erdős’s probabilistic method with high-dimensional geometry. While Erdős' original 1947 technique used randomness to prove the existence of certain mathematical objects, it struggled to provide better estimates for specific graph structures over eight decades. Researchers Wujie Shen, Jie Ma, and Shengjie Xie overcame this by placing nodes on a high-dimensional sphere and coloring edges based on distance, leveraging unique geometric properties to achieve more precise lower bounds.
* The probabilistic method proves existence through probability rather than direct construction.
* Ramsey numbers measure the threshold at which certain patterns must emerge in colored graphs.
* New research integrates geometry into random models to improve estimates for near-diagonal Ramsey numbers.
* The probabilistic method proves existence through probability rather than direct construction.
* Ramsey numbers measure the threshold at which certain patterns must emerge in colored graphs.
* New research integrates geometry into random models to improve estimates for near-diagonal Ramsey numbers.
Mathematicians are making progress on a decades-old problem about the Fourier transform by using techniques from graph theory, revealing unexpected connections between these fields.
A connection between descriptive set theory and computer science has been discovered, allowing problems in one field to be rewritten and solved in the other by Anton Bernshteyn.
Problems in descriptive set theory (measuring infinite graph colorings) are mathematically equivalent to problems in distributed algorithms (efficient network coloring).
Problems in descriptive set theory (measuring infinite graph colorings) are mathematically equivalent to problems in distributed algorithms (efficient network coloring).
Descriptive set theorists study the niche mathematics of infinity. Now, they’ve shown that their problems can be rewritten in the concrete language of algorithms.
A new mathematical proof resolves a 35-year-old bet between Noga Alon and Peter Sarnak regarding the prevalence of optimal expander graphs, demonstrating that both mathematicians were partially incorrect. The proof, building on work in random matrix theory, reveals that approximately 69% of regular graphs are Ramanujan graphs.
