Exact Simonovits product conjecture in extremal graph theory
Follow the latest progress by email.
Does a fixed forbidden graph family with polynomially superlinear extremal surplus force every extremal graph to have the exact join structure predicted by Simonovits’ Product Conjecture?
Why it matters and what counts as progress
Why it matters: Exact product structure is stronger than asymptotic descriptions of nearly extremal graphs.
What counts as progress: A structural theorem for every extremizer or a fixed forbidden family with the required surplus and extremizers whose connected complements exclude the predicted join decomposition.
How the problem developed
- 2013 · Formulation [1]
Zoltán Füredi and Miklós Simonovits Surveyed the Simonovits product-structure conjecture and its role in extremal graph theory.
- 3 Aug 2026 · Counterexample [2]
Chuandong Xu Constructed a fixed finite forbidden family with p = 2, a superlinear surplus above the Turán number, and a non-join extremal graph at every sufficiently large order.
AI contribution
A finite forbidden family with superlinear surplus and non-join extremal graphs
Xu constructs a fixed finite forbidden graph family with chromatic parameter p=2 and extremal surplus greater than c n^(3/2) for all sufficiently large n, yet with an extremal graph at each such order whose complement is connected.
Sources
Website timeline
Publication dates and editorial changes.
AI-origin graph construction refutes the exact Simonovits product conjecture