Existence of nonconstant continuous or smooth finitely forcible kernels

Does there exist a nonconstant continuous, or even smooth, symmetric kernel $W\colon[0,1]^2\to\mathbb{R}$ that is finitely forcible? More precisely, do there exist such a kernel $W$ and finitely many finite simple graphs $H_1,\ldots,H_m$ such that every bounded symmetric real-valued kernel $U\colon[0,1]^2\to\mathbb{R}$ satisfying $t(H_i,U)=t(H_i,W)$ for all $1\leq i\leq m$ is equivalent to $W$ in the usual graphon sense?

References

Primary source

arXiv

Additional references

Real-analytic finitely forcible kernels — arXiv

extremal-combinatorics graph-limits

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new preprint claims to settle the existence question by constructing nonconstant real-analytic examples, but the result has not yet been independently verified.

The problem asks whether nonconstant finitely forcible kernels exist with continuous or smooth regularity. The latest claim reaches the stronger real-analytic class.

August 2026 preprint

For every $0<\lambda\le 1/128$ , the preprint claims a nonconstant real-analytic kernel forced by a single finite graph family independent of $\lambda$ . If correct, this settles existence in the strongest stated regularity class and therefore also settles the continuous and smooth cases.

Current status (as of August 2026): Existence is claimed settled by a real-analytic construction, but the preprint remains unrefereed and the result is unverified.

Sources

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