Liu 2026 number 4 9 is non jump

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Xizhi Liu, Dhruv Mubayi, The number 4/9 is a non-jump for 3-graphs. arXiv preprint (2026). arXiv:2605.13567. The copy read for this card is arXiv:2605.13567v1 [math.CO] (13 May 2026; 12 pages). The arXiv record (https://arxiv.org/abs/2605.13567, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Theorem 1.1 (p. 2) proves that 4/9 is a non-jump for 3-uniform hypergraphs, the smallest non-jump obtained so far and below Shaw's barrier of (6/121)(5 sqrt 5 - 2) = 0.4552... for the finite-pattern formulation of the Frankl-Rodl method. The construction perturbs the ABB pattern (normalized Lagrangian max 3ab^2 = 4/9) by inserting into the B-part the union of two edge-disjoint Steiner triple systems forming a high-cogirth pair, whose existence follows from Delcourt and Postle's Theorem 3.3; the new technical ingredient is Theorem 3.1, a local Lagrangian bound showing lambda(cone(Q)) <= 4/9 for every sparse 3-graph Q with maximum codegree at most two, which replaces the finite-pattern identity lambda(FR_v(P)) = lambda(P). Via a result of Peng, Theorem 1.1 implies 2r!/r^r is a non-jump for every r >= 4, a conclusion the paper notes Shaw also obtained. Conjecture 1.2 (p. 3) boldly asserts all numbers in [0,4/9) are jumps and all in [4/9,1) are non-jumps, which would make 4/9 the smallest non-jump and answer Erdős's prize jumping-constant question in strong form. For Problem 837 this is direct progress: a new, smaller explicit non-jump value for 3-graphs.

Source: https://arxiv.org/abs/2605.13567.

Bears on. #837

Results to transcribe.

• Theorem 1.1 (p. 2): 4/9 is a non-jump for 3-uniform hypergraphs; via Peng's lifting, 2r!/r^r is a non-jump for every r >= 4 (also obtained by Shaw).
• Theorem 3.1 (p. 4): If Q is a sparse 3-graph with maximum codegree at most 2, then lambda(cone(Q)) <= 4/9.
• Proposition 3.3 (p. 4): For sparse Q with codegree at most 2 and every probability vector z on V(Q), q_Q(z) <= tau(rho(z)), which with Lemma 3.2 implies Theorem 3.1.
• Lemma 3.4 (p. 4): Universal bound q_Q(z) <= 1/27 for every sparse 3-graph Q, using sparsity alone.
• Conjecture 1.2 (p. 3): Conjectures that every number in [0,4/9) is a jump and every number in [4/9,1) is a non-jump.