Pith. sign in

REVIEW 8 cited by

The existence of designs

Not yet reviewed by Pith; the record is open.

Re-run · record.json Download PDF Read on arXiv ↗

This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.

SPECIMEN: schema-true, not a live event

T0 review · schema-true

One-sentence machine reading of the paper's core claim.

pith:XXXXXXXX · record.json · timestamp

arxiv 1401.3665 v4 pith:45CT4U6C submitted 2014-01-15 math.CO

Peter Keevash This is my paper · ORCID
classification math.CO
keywords existencecliquedesignsansweringassumingcertaincombinatorialcomplexes
verification ladder T0 review T1 audit T2 compute T3 formal
0 comments
read the original abstract

We prove the existence conjecture for combinatorial designs, answering a question of Steiner from 1853. More generally, we show that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition. As a further generalisation, we obtain the same conclusion only assuming an extendability property and the existence of a robust fractional clique decomposition.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 8 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Erd\H{o}s meets Nash-Williams

    math.CO 2025-07 conditional novelty 8.0 of 10

    Every sufficiently large triangle-divisible graph with minimum degree at least (7+√21)/14 + epsilon has a triangle decomposition with arbitrarily large girth.

  2. Relative discrepancy of hypergraphs

    math.CO 2025-06 conditional novelty 8.0 of 10

    The paper proves bs(k) ≤ g(k)+2 for k-uniform hypergraphs, determines bs(k)=3 for 3≤k≤13, and improves the known upper bound from k+1 to O(k^{0.525}).

  3. Almost every Latin square has a decomposition into transversals

    math.CO 2025-01 accept novelty 8.0 of 10

    With high probability, a uniformly random Latin square of order n has a decomposition into n disjoint transversals.

  4. Perfect Matchings in Random Sparsifications of Dense Hypergraphs

    math.CO 2025-07 conditional novelty 7.0 of 10

    A polynomial-time algorithm almost surely decides whether a random sparsification of a dense k-graph has a perfect matching, and if one exists there are exponentially many.

  5. Linear Tur\'an Numbers of Uniform Hypertrees

    math.CO 2026-07 conditional novelty 6.0 of 10

    For several r-uniform linear hypertrees with four edges, the maximum number of edges in a linear r-uniform hypergraph avoiding them is determined; the 4-uniform 4-edge path case is settled exactly.

  6. A remark on the $t$-intersecting Erd\H{o}s-Ko-Rado theorem

    math.CO 2025-07 reject novelty 5.0 of 10

    The paper attempts to show S(n,k,t)=Ω(n,k,t), but the proof as written contains false binomial identities and fails on small examples.

  7. Delta-system method: a survey

    math.CO 2025-08 conditional novelty 3.0 of 10

    A survey of the Delta-system (sunflower) method in extremal set theory, with proofs of key theorems and a broad literature review.

  8. Entropy methods in combinatorics

    math.CO 2026-07 accept novelty 2.0 of 10

    A selective survey of entropy methods in combinatorics, detailing randomized chain rules, Shearer's inequality, random homomorphisms, Pinsker-type arguments, the union-closed sets breakthrough, and entropy approaches ...

Pith tools