1 · The essentials
Two predictions about how patterns appear in networks fail for the same example. The pattern has 35 points joined by 66 links, and it is connected, so every point can be reached from every other by following links. Its points can also be split into two groups so that every link runs between the groups. Sidorenko’s conjecture predicted that such patterns should appear at least as often as in a random network with the same overall link density, using a particular mathematical way of counting. The main result gives a finite network in which this pattern falls below that prediction. The same pattern also defeats a second prediction: matching both the overall link density and this pattern’s frequency need not make a network behave like a random one.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine checking a patterned fabric by measuring both the amount of thread and the frequency of one particular motif. Those two measurements might match a randomly woven reference fabric even though the fabric has a different overall structure; the network result says that the two specified density measurements likewise need not certify randomness.
Why it matters
The counterexample rules out Sidorenko’s proposed lower bound as a statement about all bipartite patterns. It also shows that, for this same connected pattern, matching edge and pattern densities is not enough to guarantee quasirandomness. These are related but distinct failures: one concerns a minimum frequency, while the other concerns what density measurements can tell us about structure.
2 · Going further
Sidorenko’s conjecture concerns the homomorphism density of a bipartite graph in a host graph . This density counts maps from the vertices of to those of that preserve edges, normalized by the number of all vertex maps; distinct vertices need not have distinct images. If the host has edge density , the random benchmark for a pattern with 66 edges is . The main manuscript supplies a connected bipartite with 35 vertices and 66 edges and a finite simple host graph for which .
The same is also a counterexample to the forcing conjecture of Skokan and Thoma. At one fixed density, asymptotically matching both the edge density and the -density of the random benchmark does not imply quasirandomness. The repository summary expresses this in graphon language: matching these two densities with those of a constant graphon need not force quasirandomness.
The argument described in the supplied abstract is a counterexample strategy: exhibit a pattern and host violating the lower bound, and use the same pattern to defeat the forcing implication. The abstract does not give the construction or the steps connecting these two conclusions, so a more detailed mechanism cannot be extracted from the supplied text. Only one manuscript is supplied, with both conclusions attributed to the same pattern; no companion-manuscript relationship is specified.
Useful notions
- Graphs as vertices joined by edges
- Bipartite and connected graphs
- Edge density
- Edge-preserving vertex maps
- Random graph benchmarks
What was known before / what is new
Before
The source identifies Sidorenko’s conjecture as the proposed lower bound for bipartite pattern densities relative to a random graph of the same edge density. It also identifies the forcing conjecture of Skokan and Thoma, under which matching the relevant edge and pattern densities would imply quasirandomness. No further historical context is supplied.
New
The main result gives a counterexample to Sidorenko’s conjecture using a connected bipartite graph with 35 vertices and 66 edges. The manuscript also reports that this same pattern defeats the forcing conjecture at one fixed density. The main result has a Lean formalization, although the supplied material does not specify its exact coverage.
Limits and assumptions
The Sidorenko counterexample uses a particular connected bipartite pattern and some finite simple host graph; it does not assert failure for every bipartite pattern. The forcing conclusion concerns asymptotic density matching at one fixed density, not failure at every density. A Lean formalization exists for the main result, but its precise scope is not supplied, so the forcing conclusion should not automatically be treated as formally verified. The supplied abstract gives neither the explicit construction nor details of the argument, and it specifies no remaining open questions.
Mini glossary
- Bipartite graph —
- A graph whose vertices can be divided into two groups with every edge joining vertices in different groups.
- Connected graph —
- A graph in which any vertex can be reached from any other by following edges.
- Homomorphism density —
- The fraction of all vertex maps from a pattern to a host graph that preserve every edge, allowing different pattern vertices to share an image.
- Quasirandomness —
- Random-like graph behavior that the forcing conjecture seeks to infer from matching specified densities.
- Constant graphon —
- A graph model assigning the same connection value to every pair of positions, serving here as the uniform random benchmark.
Keywords : Sidorenko’s conjecture · Homomorphism density · Quasirandomness
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A counterexample to Sidorenko's conjecture
23/09/2026We disprove Sidorenko's conjecture with a bipartite graph on 35 vertices and 66 edges: its homomorphism density in some finite simple graph is smaller than the conjectured lower bound. The same connected graph also disproves the forcing conjecture: at one fixed density, asymptotically matching the edge and pattern densities does not imply quasirandomness.