Erdős Problem #1177 — Omitting finite triple systems at uncountable chromatic number

About 27 years old · traced to 1999

For a finite triple system $S$ , consider triple systems that omit $S$ and have uncountable chromatic number. (a) If one exists, must one exist with cardinality at most $2^{2^{\aleph_0}}$ ? (b) If finite triple systems $S_1$ and $S_2$ can each be omitted by an uncountably chromatic triple system, must some uncountably chromatic triple system omit both? (c) If a $\kappa$ -chromatic triple system omitting $S$ exists for some uncountable cardinal $\kappa$ , must one exist for every uncountable cardinal $\lambda$ ?

References

Primary source

Some of Paul's favorite problems, problem booklet circulated at Paul Erdős and his mathematics, Budapest, July 1999.

Additional references

Some of Paul's favorite problems, problem booklet circulated at Paul Erdős and his mathematics, Budapest, July 1999.

InErdős Problems

Posted 2 months ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2026 preprint claims that every finite forbidden triple system has either no uncountable exact chromatic sizes or all of them, but the claim is not yet independently confirmed.

Erdős Problem #1177 asks which exact uncountable chromatic cardinalities can occur for systems avoiding a fixed finite triple system, and how different avoidance classes intersect. The recent formulation concerns exact chromatic number $\aleph_1$ and extends to all uncountable cardinals.

2026 claimed resolution

A preprint claims the complete dichotomy

\operatorname{Spec}(F)=\varnothing\quad\text{or}\quad\operatorname{Spec}(F)=\{\lambda:\lambda>\aleph_0\}.

It reports the three clauses of Problem #1177 as yes, no, and yes: a witness at $\aleph_1$ can have size at most $2^{2^{\aleph_0}}$ ; two specified forbidden systems—two triples sharing a pair and the loose $7$ -cycle—have nonempty individual classes but empty intersection; and existence at one uncountable cardinal implies existence at every uncountable cardinal. Version 2 states that the results were formally verified in Lean 4.

Current status (as of June 2026): A preprint claims to settle the exact-spectrum classification and all three clauses, but independent mathematical confirmation is still lacking.

Sources

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