13 problems

Linear-hypergraph surplus conjecture. The surplus should satisfy

Berge–Füredi conjecture. A linear (loopless) hypergraph $(V,E)$ satisfies

The folklore linear-3-graph conjecture. Every linear $3$-uniform hypergraph $H$ satisfies

A $3$-graph $F$ is linear if any two of its edges share at most one vertex. For fixed $F$, let $r(F,K_n^{(3)})$ be the least $N$ such that every red-blue coloring of $K_N^{(3)}$ co…

For a triple system $F$, let $m_k(F)$ be the minimum number of edges in an $F$-free triple system with chromatic number at least $k$. Bohman–Frieze–Mubayi conjecture. There exists…

A 3-uniform hypergraph is linear if any two of its edges intersect in at most one vertex. Linear-subhypergraph conjecture. There is a constant $c>0$ such that for every integer…

Let $F$ be an $r$-uniform hypergraph and let $k$ be the uniformity of the clique $K_{k+1}^k$. Distinct edges of $F$ satisfy the $k$-linear condition when $|e\cap f|<k$ for every pa…

For a positive integer $t$, let $f(n;t)$ be the maximum number of edges in an $n$-vertex linear $3$-graph with no $2$-regular subhypergraph having at most $t$ edges. Dellamonica et…

Dellamonica et al.'s conjecture. The lower bound is asymptotically correct, namely

Defective-colouring conjecture. Every $(r+1)$-uniform linear hypergraph with maximum degree at most $\Delta$ has a $d$-defective colouring with

Dual Erdős–Faber–Lovász conjecture. Any linear hypergraph on $n$ vertices has chromatic index at most $n$.

The conjecture on larger path lengths. The exact result should hold for much larger $\ell$, possibly as far as

The conjecture on linear paths in triple systems. A similar result to the stated exact formulas for $k\geq 4$ should hold for $k=3$: for positive integers $t$, the corresponding ex…