Koishichan 2025 counterexample erdos 1022

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counterexample: Constructs a non-two-colorable uniform hypergraph whose induced edge count is at most twice its vertex count.

koishichan_2025_counterexample_erdos_1022: Records the post, attribution, date, and named acceptance of the direct counterexample to Problem 1022.

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KoishiChan, “This problem seems to admit a trivial counterexample showing that $c_t<2$ for every $t$,” comment on the Erdős Problems discussion for Problem 1022, 4 December 2025.

The construction associates two edges with each of two types of new vertices. A coloring argument shows that the resulting $(t+1)$-uniform hypergraph has no property B, while mapping every edge to its associated new vertex gives at most $2|X|$ edges inside any vertex set $X$. It follows that no constant $c>2$ can have the proposed property, which is enough to refute a sequence $c_t$ tending to infinity.

This forum result meets the repository's acceptance rule. Terence Tao replied that the argument was essentially correct and corrected its numerical conclusion to $c_t\leq2$ for this construction. Thomas Bloom then stated that Bloom would mark the problem solved. The site's problem commentary subsequently incorporated the counterexample and also cited Wood's earlier published construction.

Source. Forum source record.

Result. [[set_systems/koishichan_2025_counterexample_erdos_1022/counterexample|Direct two-level counterexample]].

Bears on. #1022