기초과학연구원 이산수학그룹
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Finding the smallest integer $N=ES_d(n)$ such that in every configuration of $N$ points in $\mathbb{R}^d$ in general position, there exist $n$ points in convex position is one of the most classical problems in extremal combinatorics, known as the Erdős-Szekeres problem. In 1935, Erdős and Szekeres famously conjectured that $ES_2(n)=2^{n−2}+1$ holds, which was nearly settled by … |
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The disjoint paths logic, FOL+DP, is an extension of First Order Logic (FOL) with the extra atomic predicate $\mathsf{dp}_k(x_1,y_1,\ldots,x_k,y_k),$ expressing the existence of internally vertex-disjoint paths between $x_i$ and $y_i,$ for $i\in \{1,\ldots, k\}$. This logic can express a wide variety of problems that escape the expressibility potential of FOL. We prove that for every … |
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Fix $r \ge 2$ and consider a family F of $C_{2r+1}$-free graphs, each having minimum degree linear in its number of vertices. Such a family is known to have bounded chromatic number; equivalently, each graph in F is homomorphic to a complete graph of bounded size. We disprove the analogous statement for homomorphic images that … |
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Various types of independent sets have been studied for decades. As an example, the minimum number of maximal independent sets in a connected graph of given order is easy to determine (hint; the answer is written in the stars). When considering this question for twin-free graphs, it becomes less trivial and one discovers some surprising … |
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