Pith Number

pith:CYDDSQJ5

pith:2026:CYDDSQJ5AB2WJSDSEBSNAN562N

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Combinatorics of Schur ultrafilters

S. Bardyla

Schur ultrafilters on countable commutative groups have a combinatorial characterization that permits constructing a free non-infinitary example on the integers and a free Schur P-point under the continuum hypothesis.

arxiv:2605.17411 v1 · 2026-05-17 · math.LO · math.CO

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Claims

C1strongest claim

We provide a combinatorial characterization of the elements of Schur ultrafilters on countable commutative groups. Using this characterization, we construct a free Schur ultrafilter on Z that is not infinitary Schur. Moreover, assuming the Continuum Hypothesis, we establish the existence of a free Schur P-point on Z.

C2weakest assumption

The Continuum Hypothesis is assumed to obtain the free Schur P-point on Z; the combinatorial characterization itself is taken as the basis for the explicit construction on Z.

C3one line summary

Combinatorial characterization of Schur ultrafilters on countable commutative groups, plus construction of a free non-infinitary Schur ultrafilter on Z and existence of a free Schur P-point on Z under CH.

References

9 extracted · 9 resolved · 1 Pith anchors

[1] S. Bardyla, P. Zlatoˇ s,Schur ultrafilters and Bohr compactifications of topological groups, Israel J. Math. accepted (2025), arXiv.2409.07280 2025

[2] D. Chodounsk´ y, O. Guzm´ an,There are no P-points in Silver extensions,Israel J. Math.232(2019), 759–773 2019

[3] Schur Number Five · arXiv:1711.08076

[4] N. Hindman, D. Strauss,Algebra in the Stone- ˇCech Compactification: Theory and Applications, Berlin, Boston: De Gruyter, 2012 2012

[5] Kunen,Weak P-points inN ∗, Topology, Vol 1978

Formal links

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Receipt and verification

First computed	2026-05-20T00:03:57.108542Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

160639413d007564c8722064d037bed3405ac9ef4d8b498096449136c42fe738

Aliases

arxiv: 2605.17411 · arxiv_version: 2605.17411v1 · doi: 10.48550/arxiv.2605.17411 · pith_short_12: CYDDSQJ5AB2W · pith_short_16: CYDDSQJ5AB2WJSDS · pith_short_8: CYDDSQJ5

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/CYDDSQJ5AB2WJSDSEBSNAN562N \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 160639413d007564c8722064d037bed3405ac9ef4d8b498096449136c42fe738

Canonical record JSON

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  "metadata": {
    "abstract_canon_sha256": "739b65df6b0a353ed61d7e613714c22e5c7325cd1e7f14b7c8c87f77b3ea43ac",
    "cross_cats_sorted": [
      "math.CO"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.LO",
    "submitted_at": "2026-05-17T12:14:30Z",
    "title_canon_sha256": "c71b0c9bc65c4e3b4f7fb680deaf3fb659c788203362814cdd55921d47f9d32f"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17411",
    "kind": "arxiv",
    "version": 1
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}