2 problems

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Tolerance complexes preserve Lerayness
Tolerance-Leray conjecture. Then $\mathcal{T}_{t}(K)$ is $(\eta(d+1,t+1)-1)$ -Leray.
Conjectureopenleray-number simplicial-complexes tolerance-complexesarXiv posted 2 days ago
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The representability bound for simplicial complexes with bounded Leray number
Representability bound conjecture. If $h(X)\leq d$ for $d\geq 1$ , then
Conjectureopenleray-number representability simplicial-complexesarXiv posted 2 days ago

Tolerance complexes preserve Lerayness