Nonprincipal Ramsey–Turán Families

Definitions and mathematical statement

For finite simple graphs, use the simultaneous-forbidden-family interpretation of Ramsey--Turán density. Let $\alpha(H)$ be the independence number and let $\operatorname{RT}(n,\mathcal F,\eta n)$ be the maximum number of edges of an $n$-vertex graph $H$ with $\alpha(H)\leq\eta n$ and no subgraph isomorphic to any $F\in\mathcal F$. Forbidden copies are not required to be induced. With an explicit upper-density convention, put \[ \rho(\mathcal F)=\lim_{\eta\downarrow0}\limsup_{n\to\infty} \frac{\operatorname{RT}(n,\mathcal F,\eta n)}{\binom n2}, \] using zero if there is no admissible graph. The source does not give its limit or normalization convention; replacing $\binom n2$ by $n^2$ changes all densities by the same factor and does not change the requested positivity and strict comparisons. A prescribed rate for sublinear independence would require a separate convention.

The requested assertion is existence of two finite graphs $G_1,G_2$ such that \[ 0<\rho(\{G_1,G_2\}) <\min\{\rho(\{G_1\}),\rho(\{G_2\})\}. \] The inequality $\rho(\{G_1,G_2\})\leq\min_i\rho(\{G_i\})$ follows from simultaneous avoidance; the open request is for strict improvement against both singleton families while retaining positive joint density. This is nonprincipality of a two-member family with the additional positive-density requirement.

The source calls this Ramsey--Turán density, whereas the literature-assessment sentence uses the shorter "Turán-density" wording. The small-independence constraint must not be dropped, and this is not a colored Ramsey problem in which a different forbidden graph is assigned to each edge color. Those alternatives would change the question. The precise source definition of $\rho$ still requires checking.

The supplied triage, checked 21 August 2026, reports the graph nonprincipal-family problem open, including this strengthened positive-joint-density version. It supplies references but no example, obstruction theorem or resolution; this draft makes no fresh literature assessment.

Short English statement

The target is a two-graph forbidden family whose Ramsey–Turán density is positive and strictly below either singleton density. The small-independence and simultaneous-avoidance conventions matter. The dated review records the existence question as open.

Sources

• Local UnsolvedMath registry, permanent ID 30000025, source code catalogue entry, supplied Oberwolfach Reports record; updated 16 September 2026.
• Combinatorics (2004), Oberwolfach Reports, DOI 10.4171/owr/2004/01, pp. 5--110: https://doi.org/10.4171/owr/2004/01.
• Mubayi--Pikhurko, supplied reference label: https://doi.org/10.1016/j.disc.2007.08.040.
• Supplied official report reference: https://ems.press/content/serial-article-files/45848.
• Dated literature triage checked 21 August 2026, derived from the supplied OWR curation; independently verify time-sensitive status and cited claims. Full article/theorem details and density conventions are not supplied.