Komorech 2025 non jumping densities 3 uniform hypergraphs

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proposition_1_4: States that a density alpha is a jump for r exactly when some c > 0 makes every large r-graph of density at least alpha + epsilon contain m-vertex subgraphs of density at least alpha + c, for every epsilon > 0 and m >= r.

theorem_1_5: Komorech's theorem that the density 64/81 is not a jump for 3-uniform hypergraphs, obtained as 3! times the Lagrangian 32/243 of a five-edge 3-pattern on three vertices.

theorem_1_6: Komorech's theorem that for every natural number n, with k = sqrt(3n - 2) (not necessarily an integer), the density 1 - (k/(n+k))^2 is not a jump for 3-uniform hypergraphs.

theorem_4_8: Komorech's main result: for a 3-pattern without the edge 111 that contains 122 and every 11i, and whose optimal weighting gives vertex 1 positive weight, the Frankl-Rödl construction at vertex 1 has the same Lagrangian.

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Vaughn Komorech, Non-jumping densities of 3-uniform hypergraphs. arXiv preprint (2025). arXiv:2511.07715; the copy read for this card is v2 (3 July 2026, 12 pages). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2511.07715), every other right reserved.

Komorech develops a method for producing non-jumps for r = 3 using hypergraph patterns (Section 3), building on Shaw's analysis of the Frankl-Rödl construction and relying on blow-ups and Lagrangians (Section 2). The main result, Theorem 4.8 (p. 7), shows that for a 3-pattern without the edge 111 that contains 122 and every 11i, and whose optimal weighting gives vertex 1 positive weight, the Frankl-Rödl construction at vertex 1 keeps the Lagrangian; with Theorem 4.2, taken from Shaw, this makes 3! times the Lagrangian a non-jump when the Lagrangian is below 1. Theorem 1.5 shows that 64/81 is not a jump for r = 3, and Theorem 1.6 gives a family indexed by n: for n in N and k = sqrt(3n - 2), the density 1 - (k/(n+k))^2 is not a jump for r = 3. The introduction surveys the state of the art: Erdős conjectured every alpha in [0,1) is a jump for every r, Frankl and Rödl disproved this by exhibiting non-jumps, and the paper says that the intervals of jumps for r = 3 found by Baber and Talbot are, apart from [0, r!/r^r), the only known jumps for r >= 3.

Source: https://arxiv.org/abs/2511.07715.

Read status. Claims checked for the four results below, each on its result page; the proofs were read for structure only.

Bears on. #837: Theorem 1.5 and Theorem 1.6 give densities that are not jumps for r = 3. The paper does not mention the problem; read through the sequence form of the jump property noted on Proposition 1.4, these densities are not in A_3. The paper does not determine A_3.

Results.

• Proposition 1.4 (p. 2): alpha is a jump for r iff there is c > 0 such that for every epsilon > 0 and every integer m >= r there is N such that every r-graph on n >= N vertices with at least (alpha + epsilon)binom(n,r) edges contains an m-vertex subgraph with at least (alpha + c)binom(m,r) edges.
• Theorem 1.5 (p. 2): "The density 64/81 is not a jump for r = 3."
• Theorem 1.6 (p. 2): for n in N and k = sqrt(3n - 2) (not necessarily integral), 1 - (k/(n+k))^2 is not a jump for r = 3.
• Theorem 4.8 (p. 7): for a 3-pattern P with 111 not an edge, containing 122 and every 11i, whose optimal weighting gives vertex 1 positive weight, lambda(FR_1(P)) = lambda(P).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.