We prove that every sufficiently large finite planar point set contains either four collinear points or seven points with at most one non-visible pair. More generally, we show that for every fixed graph $H$ with chromatic number at most five, or with chromatic number six and a color-critical edge, the visibility graph of every sufficiently large finite planar point set with no four collinear points contains a copy of $H$. These results extend the recent breakthrough of Bonnet (2026), guaranteeing six pairwise visible points, and come within one visibility edge of the next open case of the big-line-big-clique conjecture.
A point set is universal for planar graphs on $n$ vertices if every such graph has a straight-line drawing without crossings whose vertices belong to the set. We construct universal point sets of size $n^{1+o(1)}$, improving the previous quadratic upper bound. Our construction uses the reduction of Bannister, Cheng, De...
An edge card of a graph is obtained by deleting one edge, and a class edge-reconstruction number asks for the fewest carefully selected cards that identify the graph when its class is known. We determine the sharp universal bound for maximal planar graphs. Two selected cards always suffice, and the octahedral graph sho...
For $n\geq 4$, let $c(n)$ be the minimum number of distinct circles containing at least three points of an $n$-point set in the Euclidean plane, where the set is neither collinear nor concyclic. Put[F(n)=1+\binom{n-1}{2}-\left\lfloor\frac{n-1}{2}\right\rfloor.]We determine $c(n)$ for every $n\geq 4$: it equals $F(n)$ a...
A graph $H$ is called common if the number of monochromatic copies of $H$ in a $2$-edge-coloring of a large complete graph is asymptotically minimized by the random coloring with an equal proportion of each color. The house graph and $K_4$ with one edge subdivided once are known to be common by the flag algebra method....
Let $G$ be a finite simple graph and let $S\subseteq V(G)$. We prove that the minimum number of vertices meeting every cycle that intersects $S$ is at most the maximum number of vertices of $S$ covered by a collection of vertex-disjoint cycles. This answers a question posed by Bowler, Ghorbani, Gut, Jacobs, and Reich [...
We prove that there is an absolute constant $c>0$ such that every connected vertex-transitive graph $G$ on $n \ge 3$ vertices contains a cycle of length at least $cn$. Moreover, every such graph with sufficiently large degree $d$ contains a cycle of length at least $(1-d^{-1/100})n$. This gives the first linear bound t...
Jie Ma, Zi-Yuan Zhao· 1 citation· ⚡1
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