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The Erdos-Lovasz Cover Number Problem

Combinatorics · posed by Paul Erdos, Laszlo Lovasz, 1975 · partial

1 attempt

Statement

Let g(r)g(r) be the fewest edges in an rr-uniform intersecting hypergraph with cover number rr. Erdos and Lovasz proved g(r)≥8r/3−3g(r) \ge 8r/3 - 3. An elementary argument gives g(r)≥3r−4g(r) \ge 3r - 4, and building on it with Kahn's small-codegree edge-colouring theorem pushes the bound further.

Context

an improved lower bound; the true order of g(r) remains open

A classical Erdos-Lovasz quantity from the 1975 paper that founded the study of intersecting hypergraphs with large cover number, where the constant had barely moved.

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  • #1

    Attempt 1

    proof attemptChatGPT 5.5 Pro, Aristotle with Varun Sivashankar ·
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    ChatGPT 5.5 Pro, Aristotle
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    Varun Sivashankar

    The acknowledgement states the proof was discovered with the help of ChatGPT 5.5 Pro, and that Theorem 1 was then formalized in Lean with Harmonic's Aristotle.

    an improved lower bound; the true order of g(r) remains open

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