Localized hypergraph clique-weight conjecture

Let H\mathcal{H} be a qq-uniform hypergraph with mm edges, let \K(H)\K(\mathcal{H}) denote its collection of cliques, and let tq>i1t\geq q>i\geq 1. For each I(V(H)i)I\in\binom{V(\mathcal{H})}{i}, define x(I)qi1x(I)\geq q-i-1 by

d(I)=(x(I)iqi),d(I)=\binom{x(I)-i}{q-i},

and for each T\K(H)T\in\K(\mathcal{H}) define

x(T)=max{x(I):I(Ti)},s(T)=1(x(T)qtq).x(T)=\max\{x(I):I\in\binom{T}{i}\},\qquad s'(T)=\frac{1}{\binom{x(T)-q}{t-q}}.

Localized hypergraph clique-weight conjecture. One has

T\K(H)s(T)m(tq).\sum_{T\in\K(\mathcal{H})}s'(T)\leq\frac{m}{\binom{t}{q}}.

This is proposed as a localized version of a theorem of Kirsch and Radcliffe for hypergraphs, paralleling the paper's localized clique bounds. The supplied text gives no evidence that it has been proved or refuted.

Sources & referencesView supporting material

Primary source

Rachel Kirsch and JD Nir, “A localized approach to generalized Turán problems”, arXiv:2301.05678 (2023).

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