combinatorics

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An old theorem of Alexander Soifer's is the following: Given five points in a triangle of unit area, there must exist some three of them which form a triangle of area $1/4$ or less. It is easy to check that this is not true if "five" is replaced by "four", but can the theorem be improved in any other way? We discuss in this article two different extensions of the original result. First, we allow the value of "small", $1/4$, to vary. In particular, our main result is to show that given five points in a triangle of unit area, then there must exist some three of them determining a triangle of area $6/25$ or less. Second, we put bounds on the minimum number of small triangles determined by $n$ points in a triangle, and make a conjecture about the asymptotic right answer as $n \to \infty$. [NOTE: Conjecture 2.6 fails --- the optimal value is $3 - 2 \sqrt{2}$ not $1/6$.]

A graph homomorphism from the rooted $d$-branching tree $\phi: T^d \to H$ is said to be cold if the values of $\phi$ for vertices arbitrarily far away from the root can restrict the value of $\phi$ at the root. Warmth is a graph parameter that measures the non-existence of cold maps. We study warmth of random graphs $G(n,p)$, and for every $d \ge 1$, we exhibit a nearly-sharp threshold for the existence of cold maps. As a corollary, for $p=O(n^{-\alpha})$ warmth of $G(n,p)$ is concentrated on at most two values. As another corollary, a conjecture of Lovász relating mobility to chromatic number holds for ``almost all'' graphs. Finally, our results suggest new conjectures relating graph parameters from statistical physics with graph parameters from equivariant topology.