10 problems

Let $V=(J,(V_j)_{j\in J},d,H_d)$ be a hypergraph system, and suppose a strong triangle-removal-type result holds when $d=2$ and $J,H_d$ are arbitrary, as described above. Higher-di…

Ordered matrix removal conjecture. For every $k\times k$ binary matrix $A$ and every $\varepsilon>0$, there is a $\delta=\delta(k,\varepsilon)=\operatorname{poly}(\varepsilon)>0$ s…

Abundance equivalence conjecture. A graph $H$ is $F$-abundant if and only if there is an $F$-colouring $\sigma$ of $H$ such that $(H, \sigma)$ is $F$-abundant.

Kohayakawa–Nagle–Rödl conjecture. The $F$-removal lemma is polynomial if and only if $F$ is $k$-partite.

Arbitrary-group induced removal conjecture. For every $0 < \epsilon < 1/2$, if $A \subseteq G$ is $\epsilon$-far from bi-induced-$F$-free, then

Erdős's removal conjecture. If $G$ is $\varepsilon$-far from being $k$-colorable, then a uniformly sampled set of $C_k(\varepsilon)$ vertices spans a non-$k$-colorable subgraph wit…

Polynomial induced removal conjecture. The exponential bound in this theorem can be further improved to a polynomial one; equivalently, there should be a function…

Ordered binary matrix removal lemma. For any finite family $\mathcal{F}$ of ordered binary matrices and any $\epsilon > 0$ there exists $\delta = \delta(\mathcal{F}, \epsilon)$ suc…

Let $p$ be a prime and let $\epsilon,\delta>0$ be parameters in the arithmetic triangle removal lemma over a vector space. The lemma asserts that if a set of triangles has density…

Random hypergraph removal conjecture. For every $\delta>0$ and every integer $k\geq 3$, there exist constants $\epsilon>0$ and $C$ such that, if $p\geq Cn^{-1/k}$, then with probab…