Ramsey Property and Pathological Sets: Almost Disjointness, Independence and Other Maximal Objects
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915967593873408 |
|---|---|
| author | He, Jialiang Luo, Jintao Zhang, Shuguo |
| author_facet | He, Jialiang Luo, Jintao Zhang, Shuguo |
| contents | We show that under $\mathsf{ZF} + \mathsf{CC}_{\mathbb R}$, if the Ramsey property holds for all sets in a good pointclass $Γ$, then there is no MAD family in $Γ$, proving a long-standing conjecture made by A.R.D.\ Mathias in 1977. This also holds for $\mathcal I$-MAD families with respect to analytic ideals $\mathcal I$ including $\mathcal{ED}$, $\mathcal{ED}_{\mathrm{fin}}$, and $\finalphaα$ for all countable ordinals $α$. Under the same assumption, we show that if any one of the Baire property, Lebesgue measurability or Ramsey property holds for all sets in $Γ$, then there is no maximal independent family in $Γ$. Under the stronger assumption $\mathsf{ZF} + \mathsf{DC}_{\mathbb R}$, we further prove that if the Ramsey property holds for all sets in $Γ$, then $Γ$ contains no Vitali sets and thus no Hamel bases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26570 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ramsey Property and Pathological Sets: Almost Disjointness, Independence and Other Maximal Objects He, Jialiang Luo, Jintao Zhang, Shuguo Logic We show that under $\mathsf{ZF} + \mathsf{CC}_{\mathbb R}$, if the Ramsey property holds for all sets in a good pointclass $Γ$, then there is no MAD family in $Γ$, proving a long-standing conjecture made by A.R.D.\ Mathias in 1977. This also holds for $\mathcal I$-MAD families with respect to analytic ideals $\mathcal I$ including $\mathcal{ED}$, $\mathcal{ED}_{\mathrm{fin}}$, and $\finalphaα$ for all countable ordinals $α$. Under the same assumption, we show that if any one of the Baire property, Lebesgue measurability or Ramsey property holds for all sets in $Γ$, then there is no maximal independent family in $Γ$. Under the stronger assumption $\mathsf{ZF} + \mathsf{DC}_{\mathbb R}$, we further prove that if the Ramsey property holds for all sets in $Γ$, then $Γ$ contains no Vitali sets and thus no Hamel bases. |
| title | Ramsey Property and Pathological Sets: Almost Disjointness, Independence and Other Maximal Objects |
| topic | Logic |
| url | https://arxiv.org/abs/2604.26570 |