We consider the problem of reconfiguring non-crossing spanning trees on point sets. For a set P of n points in general position in the plane, the flip graph ℱ(P) has a vertex for each non-crossing spanning tree on P and an edge between any two spanning trees that can be transformed into each other by the exchange of a single edge (coined a flip). This flip graph has been intensively studied, lately with an emphasis on determining its diameter diam(ℱ(P)) for sets P of n points in convex position. For this case, the current best bounds are 14/9⋅n - O(1) ≤ diam(ℱ(P)) < 15/9⋅n - 3, obtained in a recent breakthrough work [Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber; SODA 2025]. The crucial tool for both the upper and lower bound are so-called conflict graphs, which the authors stated might be the key ingredient for determining the diameter (up to lower-order terms). In this paper, we pick up the concept of conflict graphs from the above-mentioned work and show that this tool is even more versatile than previously hoped. As our first main result, we use conflict graphs to show that computing the flip distance between two non-crossing spanning trees is NP-hard, even for point sets in convex position. Interestingly, the result still holds for more constrained flip operations, concretely, compatible flips (where the removed and the added edge do not cross) and rotations (where the removed and the added edge share an endpoint). Additionally, we present new insights on the diameter of the flip graph, by this directly extending the line of research from [BKUV SODA25]. Their lower bound is based on a constant-size pair of trees, one of which is of a type we refer to as stacked. We show that if one of the trees is stacked, then the lower bound is indeed optimal up to a constant term, that is, there exists a flip sequence of length at most 14/9⋅(n-1) to any other tree. Lastly, we improve the lower bound on the diameter of the flip graph ℱ(P) for n points in convex position to 11/7⋅n-o(n).

Abstract We say that a (multi)graph $$ \user2{G} = (\user2{V},\user2{E}) $$ has geometric thickness t if there exists a straight-line drawing $$ \user2{\varphi }:\user2{V} \to \mathbb{R}^{{\mathbf{2}}} $$ and a t -coloring of its edges where no two edges sharing a point in their relative interior have the same color. The Geometric Thickness problem asks whether a given multigraph has geometric thickness at most t . This problem was shown to be NP-hard for $$ \user2{t} = \mathbf{2} $$ (Durocher et al. Comput Geom 56:1–18, 2016. https://doi.org/10.1016/j.comgeo.2016.03.003 ). In this paper, we settle the computational complexity of Geometric Thickness by showing that it is $$\exists \mathbb {R}$$ -complete already for thickness 30 . Moreover, our reduction shows that the problem is $$\exists \mathbb {R}$$ -complete for 4392 -planar graphs, where a graph is k -planar if it admits a topological drawing with at most k crossings per edge. In the course of our paper we answer previous questions on geometric thickness and on other related problems, in particular that simultaneous graph embeddings of 31 edge-disjoint graphs and pseudo-segment stretchability with chromatic number 30 are $$\exists \mathbb {R}$$ -complete.

We study the common intersection of arrangements of double-wedges. We consider arrangements where double-wedges may be both bowties (which do not contain a vertical line) or hourglasses (which contain a vertical line), in contrast to earlier studies that focused on arrangements of only bowties. This generalization changes the setting drastically, in particular, with respect to all arguments involving the point-line duality. Namely, a point in the intersection of all double-wedges is equivalent to a line that stabs a set of segments S (corresponding to the bowties) while it avoids a different set of segments A (corresponding to the complement of the hourglasses). We show that in this general setting, the intersection of n double-wedges may consist of Ω( n 2 ) interior-disjoint regions. Further, we discuss Gallai-type results for arrangements of segments and anti-segments, and we provide algorithms for computing the intersection of such arrangements with worst-case optimal running time. Finally, we also prove that we can find a single intersection point in almost optimal running time, assuming that 3SUM admits no truly subquadratic-time algorithm.

In a simple drawing of a graph, any two edges intersect in at most one point (either a common endpoint or a proper crossing). A simple drawing is generalized twisted if it fulfills certain rather specific constraints on how the edges are drawn. An abstract rotation system of a graph assigns to each vertex a cyclic order of its incident edges. A realizable rotation system is one that admits a simple drawing such that at each vertex, the edges emanate in that cyclic order, and a generalized twisted rotation system can be realized as a generalized twisted drawing. Generalized twisted drawings have initially been introduced to obtain improved bounds on the size of plane substructures in any simple drawing of K_n. They have since gained independent interest due to their surprising properties. However, the definition of generalized twisted drawings is very geometric and drawing-specific. In this paper, we develop characterizations of generalized twisted drawings that enable a purely combinatorial view on these drawings and lead to efficient recognition algorithms. Concretely, we show that for any n ≥ 7, an abstract rotation system of K_n is generalized twisted if and only if all subrotation systems induced by five vertices are generalized twisted. This implies a drawing-independent and concise characterization of generalized twistedness. Besides, the result yields a simple O(n⁵)-time algorithm to decide whether an abstract rotation system is generalized twisted and sheds new light on the structural features of simple drawings. We further develop a characterization via the rotations of a pair of vertices in a drawing, which we then use to derive an O(n²)-time algorithm to decide whether a realizable rotation system is generalized twisted.

A flip in a plane spanning tree T is the operation of removing one edge from T and adding another edge such that the resulting structure is again a plane spanning tree. For trees on a set of points in convex position we study two classic types of constrained flips: (1) Compatible flips are flips in which the removed and inserted edge do not cross each other. We relevantly improve the previous upper bound of 2n-O(√n) on the diameter of the compatible flip graph to (5n/3)-O(1), by this matching the upper bound for unrestricted flips by Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber [SODA 2025] up to an additive constant of 1. We further show that no shortest compatible flip sequence removes an edge that is already in its target position. Using this so-called happy edge property, we derive a fixed-parameter tractable algorithm to compute the shortest compatible flip sequence between two given trees. (2) Rotations are flips in which the removed and inserted edge share a common vertex. Besides showing that the happy edge property does not hold for rotations, we improve the previous upper bound of 2n-O(1) for the diameter of the rotation graph to (7n/4)-O(1).

For a finite set P of points in the plane in general position, a crossing family of size k in P is a collection of k line segments with endpoints in P that are pairwise crossing. It is a long-standing open problem to determine the largest size of a crossing family in any set of n points in the plane in general position. It is widely believed that this size should be linear in n. Motivated by results from the theory of partitioning complete geometric graphs, we study a variant of this problem for point sets P that do not contain a non-crossing family of size m, which is a collection of 4 disjoint subsets P₁, P₂, P₃, and P₄ of P, each containing m points of P, such that for every choice of 4 points p_i ∈ P_i, the set {p₁,p₂,p₃,p₄} is such that p₄ is in the interior of the triangle formed by p₁,p₂,p₃. We prove that, for every m ∈ ℕ, each set P of n points in the plane in general position contains either a crossing family of size n/2^{O(√{log{m}})} or a non-crossing family of size m, by this strengthening a recent breakthrough result by Pach, Rubin, and Tardos (2021). Our proof is constructive and we show that these families can be obtained in expected time O(nm^{1+o(1)}). We also prove that a crossing family of size Ω(n/m) or a non-crossing family of size m in P can be found in expected time O(n).

A connected topological drawing of a graph divides the plane into a number of cells. The type of a cell c is the cyclic sequence of crossings and vertices along the boundary walk of c. For example, all triangular cells with three incident crossings and no incident vertex share the same cell type. When a non-homotopic drawing of an n-vertex multigraph G does not contain any such cells, Ackerman and Tardos [JCTA 2007] proved that G has at most 8n-20 edges, while Kaufmann, Klemz, Knorr, Reddy, Schröder, and Ueckerdt [GD 2024] showed that this bound is tight. In this paper, we initiate the in-depth study of non-homotopic drawings that do not contain one fixed cell type 𝔠, and investigate the edge density of the corresponding multigraphs, i.e., the maximum possible number of edges. We consider non-homotopic as well as simple drawings, multigraphs as well as simple graphs, and every possible type of cell. For every combination of drawing style, graph type, and cell type, we give upper and lower bounds on the corresponding edge density. With the exception of the cell type with four incident crossings and no incident vertex, we show for every cell type 𝔠 that the edge density of n-vertex (multi)graphs with 𝔠-free drawings is either quadratic in n or linear in n. In most cases, our bounds are tight up to an additive constant. Additionally, we improve the current lower bound on the edge density of simple graphs that admit a non-homotopic quasiplanar drawing from 7n-28 to 7.5n-28.

For a set P of n points in general position in the plane, the flip graph F (P ) has a vertex for each noncrossing spanning tree on P and an edge between any two spanning trees that can be transformed into each other by one edge flip, i.e., the deletion and addition of exactly one edge. The diameter diam(F (P )) of this flip graph is subject of intensive study. For points P in general position, it is between and 2n - 4, with no improvement for 25 years. For points P in convex position, diam(F (P )) lies between and ≈ 1.95n, where the lower bound was conjectured to be tight up to an additive constant and the upper bound is a very recent breakthrough improvement over several previous bounds of the form 2n - o (n ).

Visualizing a graph $G$ in the plane nicely, for example, without crossings, is unfortunately not always possible. To address this problem, Masařík and Hliněný [GD 2023] recently asked for each edge of $G$ to be drawn without crossings while allowing multiple different drawings of $G$. More formally, a collection $\mathcal{D}$ of drawings of $G$ is uncrossed if, for each edge $e$ of $G$, there is a drawing in $\mathcal{D}$ such that $e$ is uncrossed. The uncrossed number $\mathrm{unc}(G)$ of $G$ is then the minimum number of drawings in some uncrossed collection of $G$. No exact values of the uncrossed numbers have been determined yet, not even for simple graph classes. In this paper, we provide the exact values for uncrossed numbers of complete and complete bipartite graphs, partly confirming and partly refuting a conjecture posed by Hliněný and Masařík. We also present a strong general lower bound on $\mathrm{unc}(G)$ in terms of the number of vertices and edges of $G$. Moreover, we prove NP-hardness of the related problem of determining the edge crossing number of a graph $G$, which is the smallest number of edges of $G$ taken over all drawings of $G$ that participate in a crossing. This problem was posed as open by Schaefer in his book [Crossing Numbers of Graphs 2018].

Flip graphs of combinatorial and geometric objects are at the heart of many deep structural insights and connections between different branches of discrete mathematics and computer science. They also provide a natural framework for the study of reconfiguration problems. We study flip graphs of arrangements of pseudolines and of arrangements of pseudocircles, which are combinatorial generalizations of lines and circles, respectively. In both cases we consider triangle flips as local transformation and prove conjectures regarding their connectivity.

Abstract For sets of n points, n even, in general position in the plane, we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least $$C_{n/2}$$ C n / 2 different plane perfect matchings, where $$C_{n/2}$$ C n / 2 is the n /2-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every $$k\le \frac{1}{64}n^2-\frac{35}{32}n\sqrt{n}+\frac{1225}{64}n$$ k ≤ 1 64 n 2 - 35 32 n n + 1225 64 n , any set with n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has at most $$\frac{5}{72}n^2-\frac{n}{4}$$ 5 72 n 2 - n 4 crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n . (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for $$k=0,1,2$$ k = 0 , 1 , 2 , and maximize the number of perfect matchings with $$\left( {\begin{array}{c}n/2\\ 2\end{array}}\right) $$ n / 2 2 crossings and with $${\left( {\begin{array}{c}n/2\\ 2\end{array}}\right) }\!-\!1$$ n / 2 2 - 1

Abstract Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). A simple drawing is c-monotone if there is a point O such that each ray emanating from O crosses each edge of the drawing at most once. We introduce a special kind of c-monotone drawings that we call generalized twisted drawings. A c-monotone drawing is generalized twisted if there is a ray emanating from O that crosses all the edges of the drawing. Via this class of drawings, we show that every simple drawing of the complete graph with n vertices contains $$\Omega (n^{\frac{1}{2}})$$ Ω ( n 1 2 ) pairwise disjoint edges and a plane cycle (and hence path) of length $$\Omega (\frac{\log n }{\log \log n})$$ Ω ( log n log log n ) . Both results improve over best previously published lower bounds. On the way we show several structural results and properties of generalized twisted and c-monotone drawings, some of which we believe to be of independent interest. For example, we show that a drawing D is c-monotone if there exists a point O such that no edge of D is crossed more than once by any ray that emanates from O and passes through a vertex of D.

Abstract We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in $${\mathbb {R}}^3$$ R 3 . We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains $$K_5$$ K 5 , $$K_{5,81}$$ K 5 , 81 , or any nonplanar 3-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, $$K_{4,4}$$ K 4 , 4 , and $$K_{3,5}$$ K 3 , 5 can be realized with convex cells. The same holds for any subdivision of any graph where each edge is subdivided at least once, and, by a result from McMullen et al. (Isr. J. Math. 46(1–2), 127–144 (1983)), for any hypercube. Our results have implications on the maximum density of graphs describing polyhedral surfaces with convex cells: The realizability of hypercubes shows that the maximum number of edges over all realizable n-vertex graphs is in $$\Omega (n\log n)$$ Ω ( n log n ) . From the non-realizability of $$K_{5,81}$$ K 5 , 81 , we obtain that any realizable n-vertex graph has $${\mathcal {O}}(n^{9/5})$$ O ( n 9 / 5 ) edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.

Abstract A simple drawingD(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of $$G+e$$ G + e extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is -complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles $$\mathcal {A}$$ A and a pseudosegment $$\sigma $$ σ , it can be decided in polynomial time whether there exists a pseudocircle $$\Phi _\sigma $$ Φ σ extending $$\sigma $$ σ for which $$\mathcal {A}\cup \{\Phi _\sigma \}$$ A ∪ { Φ σ } is again an arrangement of pseudocircles.

The total angular resolution of a straight-line drawing is the minimum angle between two edges of the drawing. It combines two properties contributing to the readability of a drawing: the angular resolution, which is the minimum angle between incident edges, and the crossing resolution, which is the minimum angle between crossing edges. We consider the total angular resolution of a graph, which is the maximum total angular resolution of a straight-line drawing of this graph. We prove tight bounds for the number of edges for graphs for some values of the total angular resolution up to a finite number of well specified exceptions of constant size. In addition, we show that deciding whether a graph has total angular resolution at least 60∘ is NP-hard. Further we present some special graphs and their total angular resolution.

In this paper, we disprove the long-standing conjecture that any complete geometric graph on 2n vertices can be partitioned into n plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize which bumpy wheels can and in particular which cannot be partitioned into plane spanning trees (or even into arbitrary plane subgraphs). Furthermore, we show a sufficient condition for generalized wheels to not admit a partition into plane spanning trees, and give a complete characterization when they admit a partition into plane spanning double stars. Finally, we initiate the study of partitions into beyond planar subgraphs, namely into k-planar and k-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.

Abstract Let S be a set of n points in general position in the plane, and let $$X_{k,\ell }(S)$$ X k , ℓ ( S ) be the number of convex k-gons with vertices in S that have exactly $$\ell $$ ℓ points of S in their interior. We prove several equalities for the numbers $$X_{k,\ell }(S)$$ X k , ℓ ( S ) . This problem is related to the Erdős–Szekeres theorem. Some of the obtained equations also extend known equations for the numbers of empty convex polygons to polygons with interior points. Analogous results for higher dimension are shown as well.

We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in ℝ³. We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains K_5, K_{5,81}, or any nonplanar 3-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, K_{4,4}, and K_{3,5} can be realized with convex cells. The same holds for any subdivision of any graph where each edge is subdivided at least once, and, by a result from McMullen et al. (1983), for any hypercube. Our results have implications on the maximum density of graphs describing polyhedral surfaces with convex cells: The realizability of hypercubes shows that the maximum number of edges over all realizable n-vertex graphs is in Ω(n log n). From the non-realizability of K_{5,81}, we obtain that any realizable n-vertex graph has 𝒪(n^{9/5}) edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.

In extension problems of partial graph drawings one is given an incomplete drawing of an input graph G and is asked to complete the drawing while maintaining certain properties. A prominent area where such problems arise is that of crossing minimization. For plane drawings and various relaxations of these, there is a number of tractability as well as lower-bound results exploring the computational complexity of crossing-sensitive drawing extension problems. In contrast, comparatively few results are known on extension problems for the fundamental and broad class of simple drawings, that is, drawings in which each pair of edges intersects in at most one point. In fact, the extension problem of simple drawings has only recently been shown to be NP-hard even for inserting a single edge. In this paper we present tractability results for the crossing-sensitive extension problem of simple drawings. In particular, we show that the problem of inserting edges into a simple drawing is fixed-parameter tractable when parameterized by the number of edges to insert and an upper bound on newly created crossings. Using the same proof techniques, we are also able to answer several closely related variants of this problem, among others the extension problem for k-plane drawings. Moreover, using a different approach, we provide a single-exponential fixed-parameter algorithm for the case in which we are only trying to insert a single edge into the drawing.

Abstract Flip graphs are a ubiquitous class of graphs, which encode relations on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other? We consider flip graphs on orientations of simple graphs, where flips consist of reversing the direction of some edges. More precisely, we consider so-called $$\alpha$$ α -orientations of a graph G, in which every vertex v has a specified outdegree $$\alpha (v)$$ α ( v ) , and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two $$\alpha$$ α -orientations of a planar graph G is at most two is -complete. This also holds in the special case of perfect matchings, where flips involve alternating cycles. This problem amounts to finding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope. It therefore provides an interesting example of a flip distance question that is computationally intractable despite having a natural interpretation as a geodesic on a nicely structured combinatorial polytope. We also consider the dual question of the flip distance between graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard. However, if we restrict to flips that only change sinks into sources, or vice-versa, then the problem can be solved in polynomial time. Here we exploit the fact that the flip graph is the cover graph of a distributive lattice. This generalizes a recent result from Zhang et al. (Acta Math Sin Engl Ser 35(4):569–576, 2019).

The Harary--Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph $K_n$ is \(H(n) := \frac 1 4 łfloor\fracn2\rfloor łfloor\fracn-12\rfloor łfloor\fracn-22\rfloor łfloor\fracn-32\rfloor.\) Ábrego et al. [ Discrete Comput. Geom., 52 (2014), pp. 743--753] introduced the notion of shellability of a drawing $D$ of $K_n$. They proved that if $D$ is $s$-shellable for some $s\geq\lfloor\frac{n}{2}\rfloor$, then $D$ has at least $H(n)$ crossings. This is the first combinatorial condition on a drawing that guarantees at least $H(n)$ crossings. In this work, we generalize the concept of $s$-shellability to bishellability, where the former implies the latter in the sense that every $s$-shellable drawing is, for any $b \leq s-2$, also $b$-bishellable. Our main result is that $(\lfloor \frac{n}{2} \rfloor-2)$-bishellability of a drawing $D$ of $K_n$ also guarantees, with a simpler proof than for $s$-shellability, that $D$ has at least $H(n)$ crossings. We exhibit a drawing of $K_{11}$ that has $H(11)$ crossings, is 3-bishellable, and is not $s$-shellable for any $s\geq5$. This shows that we have properly extended the class of drawings for which the Harary--Hill conjecture is proved. Moreover, we provide an infinite family of drawings of $K_n$ that are $(\lfloor \frac{n}{2} \rfloor-2)$-bishellable, but not $s$-shellable for any $s\geq\lfloor\frac{n}{2}\rfloor$.

We consider the problem of routing a data packet through the visibility graph of a polygonal domain P with n vertices and h holes. We may preprocess P to obtain a label and a routing table for each vertex. Then, we must be able to route a data packet between any two vertices p and q of P , where each step must use only the label of the target node q and the routing table of the current node. For any fixed eps > 0, we pre ent a routing scheme that always achieves a routing path that exceeds the shortest path by a factor of at most 1 + eps. The labels have O(log n) bits, and the routing tables are of size O((eps^{-1} + h) log n). The preprocessing time is O(n^2 log n + hn^2 + eps^{-1}hn). It can be improved to O(n 2 + eps^{-1}n) for simple polygons.

Upper and lower bounds for the number of geometric graphs of specific types on a given set of points in the plane have been intensively studied in recent years. For most classes of geometric graphs it is now known that point sets in convex position minimize their number. However, it is still unclear which point sets minimize the number of geometric triangulations; the so-called double circles are conjectured to be the minimizing sets. In this paper we prove that any set of n points in general position in the plane has at least Omega(2.631^n) geometric triangulations. Our result improves the previously best general lower bound of Omega(2.43^n) and also covers the previously best lower bound of Omega(2.63^n) for a fixed number of extreme points. We achieve our bound by showing and combining several new results, which are of independent interest: (1) Adding a point on the second convex layer of a given point set (of 7 or more points) at least doubles the number of triangulations. (2) Generalized configurations of points that minimize the number of triangulations have at most n/2 points on their convex hull. (3) We provide tight lower bounds for the number of triangulations of point sets with up to 15 points. These bounds further support the double circle conjecture.

Let P = B ∪ R be a set of 2n points in general position, where B is a set of n blue points and R a set of n red points. A BR-matching is a plane geometric perfect matching on P such that each edge has one red endpoint and one blue endpoint. Two BR-matchings are compatible if their union is also plane.

Given a set B of n black points in general position, we say that a set of white points W blocks B if in the Delaunay triangulation of B ∪ W there is no edge connecting two black points. We give the following bounds for the size of the smallest set W blocking B : (i) 3 n / 2 white points are always sufficient to block a set of n black points, (ii) if B is in convex position, 5 n / 4 white points are always sufficient to block it, and (iii) at least n − 1 white points are always necessary to block a set of n black points.

We study the problem how to draw a planar graph crossing-free such that every vertex is incident to an angle greater than π . In general a plane straight-line drawing cannot guarantee this property. We present algorithms which construct such drawings with either tangent-continuous biarcs or quadratic Bézier curves (parabolic arcs), even if the positions of the vertices are predefined by a given plane straight-line drawing of the graph. Moreover, the graph can be drawn with circular arcs if the vertices can be placed arbitrarily. The topic is related to non-crossing drawings of multigraphs and vertex labeling.

We consider a variation of a problem stated by Erdős and Szekeres in 1935 about the existence of a number $f^{\mathrm{ES}}(k)$ such that any set S of at least $f^{\mathrm{ES}}(k)$ points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any sufficiently large bichromatic set of points in $\mathbb{R}^2$ in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).