Shapira 2023 new approach brown erdos sos problem

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conjecture_1_1: The constant-deficiency form of the Brown-Erdős-Sós conjecture, which the paper reduces to a Turán-type conjecture on 2-degenerate bipartite graphs.

theorem_1_4: The paper's reduction of the constant-deficiency Brown-Erdős-Sós conjecture to a weakened form of Conlon's conjecture on Turán numbers of 2-degenerate C4-free bipartite graphs, avoiding hypergraph regularity.

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Asaf Shapira, Mykhaylo Tyomkyn, A new approach for the Brown-Erdős-Sós problem. arXiv preprint (2023). arXiv:2301.07758.

The retained folder-name PDF is arXiv:2301.07758v1 (18 January 2023, 8 pages, the only arXiv version per the arXiv record read). The paper appeared as an extended abstract in the EuroComb 2023 proceedings (doi:10.5817/cz.muni.eurocomb23-112, pp. 812--818) and in Israel J. Math. 267 (2025), no. 2, 717--728, doi:10.1007/s11856-025-2714-5 (Crossref records); neither is held or compared, so their labels may differ from the preprint's. The arXiv record (https://arxiv.org/abs/2301.07758, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Read status: claims checked for Conjectures 1.1--1.3 and Theorem 1.4 with the remark after it (pp. 2--3), read clause by clause on the page images; the proof of Theorem 1.4 (Section 2, pp. 3--6) was read for structure only, and of Section 3 only the statement of Claim 3.1 (p. 6) was read, on the page image. Paged at conjecture_1_1 and theorem_1_4.

The Brown-Erdos-Sos conjecture asserts that for fixed e every 3-uniform hypergraph with Omega(n^2) edges contains e edges spanned by e + 3 vertices; it is known only for e = 3 (Ruzsa-Szemeredi), and hypergraph-regularity approaches give only e + O(log e / log log e) vertices. This short paper avoids regularity entirely and instead reduces the constant-deficiency version (Conjecture 1.1: some absolute d with (e + d, e)-configurations always present) to Conjecture 1.3, a weakening of Conlon's conjecture on Turan numbers of 2-degenerate C4-free bipartite graphs, asking for absolute constants t, k0 such that, for every k >= k0 and large n, every graph with Omega(n^{3/2}) edges contains some member of the family H_{k,t} of 2-degenerate graphs on k vertices with 2k - t edges. Claim 3.1 notes that H_{k,t} contains C4-free graphs, so Conjecture 1.2 (Conlon) implies Conjecture 1.3. For problem 1157 the paper is citation-backed evidence of active expert attention: it presents the Brown-Erdos-Sos conjecture for 3-graphs as among the most famous open problems of its kind in extremal combinatorics (p. 1) and contributes a new 2023 reduction rather than a resolution.

Source: https://arxiv.org/abs/2301.07758.

Bears on. #1157: Conjecture 1.1 (p. 2; conjecture_1_1) is the constant-deficiency weakening of the Brown-Erdős-Sós conjecture, and Theorem 1.4 (p. 3; theorem_1_4) reduces it to Conjecture 1.3, a conditional route and not a bound.

Results to transcribe.

• Conjecture 1.1 (p. 2): Constant-deficiency BESC: an absolute constant d such that every 3-graph with Omega(n^2) edges contains an (e + d, e)-configuration.
• Conjecture 1.3 (p. 2): Weakened Conlon-type conjecture: absolute constants t, k0 such that for every k >= k0 and large n every graph with Omega(n^{3/2}) edges contains a copy of some H in H_{k,t}.
• Theorem 1.4 (p. 3, the main reduction): Conjecture 1.3 implies Conjecture 1.1, so a proof of Conjecture 1.3 would resolve BESC up to an absolute additive constant; the reduction avoids hypergraph regularity.
• Claim 3.1 (p. 6): for every g there is t = t(g) such that for every k >= t some k-vertex bipartite graph of girth at least g arises from t isolated vertices by adding vertices of degree exactly 2 one at a time; the paper cites it (p. 2) for C4-free graphs in H_{k,t}, which make Conjecture 1.3 a consequence of Conlon's Conjecture 1.2.