Exact Ordered Ruzsa-Szemeredi Numbers for Matchings of Size Two

[2608.14695] Ordered Ruzsa-Szemeredi Numbers at Matching Size Two Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2608.14695 (math) [Submitted on 9 Aug 2026 ( v1 ), last revised 30 Aug 2026 (this version, v2)] Title: Ordered Ruzsa-Szemeredi Numbers at Matching Size Two Authors: Xidan Song , Ruifeng Cao View a PDF of the paper titled Ordered Ruzsa-Szemeredi Numbers at Matching Size Two, by Xidan Song and 1 other authors View PDF HTML (experimental) Abstract: Bondy and Szwarcfiter defined $\mathrm{ex}^*(n,F)$ as the largest number of edges in an $n$-vertex graph whose edge set partitions into induced copies of $F$; for $F=2K_2$ the deficiency $\binom{n}{2}-\mathrm{ex}^*(n,2K_2)$ is $\Theta(n^{3/2})$. We study the ordered relaxation at fixed matching size, in which each part need only be induced in the union of itself with the parts that follow it; write $\mathrm{ORS}_n(r)$ for the largest number of parts, so that $r\,\mathrm{ORS}_n(r)$ is the ordered analogue of $\mathrm{ex}^*(n,rK_2)$. Our main tool is a characterisation valid for every $r$: an ordered decomposition into induced $r$-matchings is a sequence of steps that start from $K_n$ and repeatedly delete a perfect matching from $2r$ vertices currently spanning a clique. Reading a decomposition backwards turns a condition about the ordering into a reachability question that an exhaustive search can settle. For $r=2$ we determine $\mathrm{ORS}_n(2)$ exactly at orders five through nineteen, where it takes the values $1,3,5,8,11,14,19,23,28,34,40,47,54,62,70$, and we confine $\mathrm{ORS}_{20}(2)$ to $\{78,79\}$. The counting bound $\lfloor n(n-4)/4\rfloor$ is attained at orders five through nine and at eleven, and missed by exactly one part at every other order below twenty, so order eleven is an isolated exception, not a parity effect. Across this range the ordered deficiency equals $\frac32n+O(1)$, and along powers of two a dyadic construction keeps it below $O(n\log n)$; whether it is linear for all $n$ is our main open question. The structural results are formalised in Lean 4, and the searches are certified by fail-closed sweeps and an independent checker. Comments: 26 pages, 2 figures. v2: substantially revised and extended. Ordered/unordered deficiency comparison added; exact values determined through order nineteen with order twenty confined to two values; Lean 4 formalisation of the structural results included, with a verified build; supplementary code and certificates included Subjects: Combinatorics (math.CO) ; Discrete Mathematics (cs.DM) MSC classes: 05C35, 05C70, 05C85 Cite as: arXiv:2608.14695 [math.CO] (or arXiv:2608.14695v2 [math.CO] for this version) https://doi.org/10.48550/arXiv.2608.14695 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Ruifeng Cao [ view email ] [v1] Sun, 9 Aug 2026 20:41:29 UTC (7,544 KB) [v2] Sun, 30 Aug 2026 11:22:17 UTC (7,557 KB) Full-text links: Access Paper: View a PDF of the paper titled Ordered Ruzsa-Szemeredi Numbers at Matching Size Two, by Xidan Song and 1 other authors View PDF HTML (experimental) TeX Source view license Ancillary-file links: Ancillary files ( details ) : README.txt code/README.txt code/SHA256SUMS.txt code/VERIFY_ALL.py code/c16_upper_bound_evidence.py code/cubic20_edge_reducible_enum_s62.py code/cubic20_enum_fast_s62.cpp code/cubic20_find_diamond_gap_s62.cpp code/cubic20_triangle_gap_s62.cpp code/enumerate_base_cases.py code/ors17_lb54_s61.cpp code/ors17_lb54_s61.py code/ors18_lb62_s61.cpp code/ors18_lb62_s61.py code/ors19_lb70_s61.cpp code/ors19_lb70_s61.py code/ors20_crosscheck_s62.py code/ors20_lb78_s62.cpp code/ors20_lb78_s62.json code/ors20_lb78_source_rem77.txt code/ors20_upper_s62.cpp code/orslib/__init__.py code/orslib/buildup.py code/orslib/canon.py code/orslib/core.py code/orslib/graphs.py code/orslib/solver_complete.py code/results/c16_doubly_irreducible.json code/results/cubic10_classes.json code/results/cubic12_classes.json code/results/cubic14_classes.json code/results/cubic16_classes.json code/results/cubic18_doubly_irreducible.json code/results/cubic18_sweep_state.json code/results/cubic20_census_s62.json code/results/cubic20_edge_reducible_full_s62.out code/results/cubic20_edge_reducible_s62.bin code/results/cubic20_gap4_s62.bin code/results/cubic20_gap4_s62.json code/results/cubic20_gap4_validation_s62.json code/results/cubic20_gap_discovery_s62.json code/results/cubic20_triangle_gap_s62.out code/results/nearcubic13_classes.json code/results/ors11_witness.json code/results/ors12_witness.json code/results/ors13_witness.json code/results/ors14_witness.json code/results/ors15_witness.json code/results/ors16_witness.json code/results/ors17_lb54_s61.json code/results/ors17_lb_peel.json code/results/ors17_upper_s60.json code/results/ors18_lb62_s61.json code/results/ors18_lb_peel.json code/results/ors18_source_reaudit_s60.json code/results/ors19_lb70_s61.json code/results/ors19_upper_s61.json code/results/ors20_cpp_prefix1000_s62.out code/results/ors20_crosscheck_1000_s62.json code/results/ors20_upper_s62.json code/results/ors20_upper_s62.out code/results/ors2_full_verification_s58.json code/results/ors2_machine_manifest.json code/results/ors2_witness_verification_s58.json code/results/tables.json code/results/verify_ors20_upper_s62_O.json code/results/verify_ors20_upper_s62_OO.json code/results/verify_ors20_upper_s62_full.json code/results/verify_ors20_upper_s62_normal.json code/verify_cubic20_gap4_s62.py code/verify_ors17_upper.py code/verify_ors19_upper_s61.cpp code/verify_ors19_upper_s61.py code/verify_ors20_lb78_s62.py code/verify_ors20_upper_s62.py code/verify_ors2_machine.py code/verify_ors2_witnesses.py lean/K4Peeling.lean lean/K4Peeling/Axioms.lean lean/K4Peeling/Basic.lean lean/K4Peeling/Connectivity.lean lean/K4Peeling/Counting.lean lean/K4Peeling/Local.lean lean/K4Peeling/ORS/Bijection.lean lean/K4Peeling/ORS/Defs.lean lean/K4Peeling/ORS/Main.lean lean/K4Peeling/Peeling.lean lean/K4Peeling/Structure.lean lean/README.md lean/build_evidence/README.txt lean/build_evidence/axioms.log lean/build_evidence/build.log lean/build_evidence/versions.log lean/lake-manifest.json lean/lakefile.toml lean/lean-toolchain (91 additional files not shown) Current browse context: math.CO < prev | next > new | recent | 2026-08 Change to browse by: cs cs.DM math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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