## Periodic Reporting for period 1 - GeoScape (From Geometry to Combinatorics and Back: Escaping the Curse of Dimensionality)

Reporting period: 2020-09-01 to 2022-02-28

Our main objective was to study hard combinatorial problems for large classes of graphs and hypergraphs arising in geometric,

algebraic, and practical applications. In particular, for structures escaping the

“curse of dimensionality”: if they can be embedded in a bounded-dimensional

space, or have a short algebraic description.

algebraic, and practical applications. In particular, for structures escaping the

“curse of dimensionality”: if they can be embedded in a bounded-dimensional

space, or have a short algebraic description.

The disjointness graph G of a set of segments S is a graph whose vertex set is S and two vertices are adjacent if and only if the corresponding segments are disjoint.

We proved the chromatic number of such G is polynomially bounded by the clique number of G.

We showed that computing the clique number and the chromatic number for disjointness graphs of lines in space are NP-hard tasks.

We constructed families of arcs, whose disjointness graphs are triangle-free, but whose chromatic numbers are arbitrarily large.

Let G be a multigraph drawn in the plane such that any two parallel edges form a simple closed curve with at least one vertex in its interior and at least one vertex in its exterior.

Pach and Tóth extended the Crossing Lemma of Ajtai et al. and Leighton by showing that if no two adjacent edges cross and every pair of nonadjacent edges

cross at most once, then the number of edge crossings in G is the same order as the Crossing Lemma.

The situation turns out to be quite different if nonparallel edges are allowed to cross any number of times.

We proved in this case the number of crossings in G has a different order of magnitude.

A “cornerstone” of the proposed research was to approach notoriously difficult unsolved problems for bounded-dimensional set systems.

We successfully used this idea to prove a slightly weaker version the famous Sunflower Conjecture of Erdős and Rado.

We came close to proving this conjecture for families of bounded Vapnik-Chervonenkis dimension.

We also verified the Erdős-Rado conjecture for families F of bounded Littlestone dimension and for some geometrically defined set systems.

For n ≤ d, a family F of n+1 compact convex sets in the d-dimensional real space is an n-critical family if any n members of F have non-empty intersection, but all sets have empty intersection.

If n = d, then a lemma on the intersection of 3 convex sets due to Klee implies that the d + 1 members of the d-critical family enclose a “hollow”, a bounded connected component.

We proved the closure of the convex hull of a hollow is a d-simplex.

We developed a theory for the convergence and limits of finite trees. This concept is similar

to the celebrated limit theory of finite graphs by Lovász, Szegedy and many

others. We worked with random samples of vertices. However, since trees are sparse,

therefore the induced subgraphs are not informative (they are almost always empty).

To get around this problem, we considered trees as metric spaces with a normalized version

of graph distance. This way one can apply the well-developed theory of metric

measures. Surprisingly, the limit objects (which we call dendrons) are typically not metric

measure spaces.

We proved the chromatic number of such G is polynomially bounded by the clique number of G.

We showed that computing the clique number and the chromatic number for disjointness graphs of lines in space are NP-hard tasks.

We constructed families of arcs, whose disjointness graphs are triangle-free, but whose chromatic numbers are arbitrarily large.

Let G be a multigraph drawn in the plane such that any two parallel edges form a simple closed curve with at least one vertex in its interior and at least one vertex in its exterior.

Pach and Tóth extended the Crossing Lemma of Ajtai et al. and Leighton by showing that if no two adjacent edges cross and every pair of nonadjacent edges

cross at most once, then the number of edge crossings in G is the same order as the Crossing Lemma.

The situation turns out to be quite different if nonparallel edges are allowed to cross any number of times.

We proved in this case the number of crossings in G has a different order of magnitude.

A “cornerstone” of the proposed research was to approach notoriously difficult unsolved problems for bounded-dimensional set systems.

We successfully used this idea to prove a slightly weaker version the famous Sunflower Conjecture of Erdős and Rado.

We came close to proving this conjecture for families of bounded Vapnik-Chervonenkis dimension.

We also verified the Erdős-Rado conjecture for families F of bounded Littlestone dimension and for some geometrically defined set systems.

For n ≤ d, a family F of n+1 compact convex sets in the d-dimensional real space is an n-critical family if any n members of F have non-empty intersection, but all sets have empty intersection.

If n = d, then a lemma on the intersection of 3 convex sets due to Klee implies that the d + 1 members of the d-critical family enclose a “hollow”, a bounded connected component.

We proved the closure of the convex hull of a hollow is a d-simplex.

We developed a theory for the convergence and limits of finite trees. This concept is similar

to the celebrated limit theory of finite graphs by Lovász, Szegedy and many

others. We worked with random samples of vertices. However, since trees are sparse,

therefore the induced subgraphs are not informative (they are almost always empty).

To get around this problem, we considered trees as metric spaces with a normalized version

of graph distance. This way one can apply the well-developed theory of metric

measures. Surprisingly, the limit objects (which we call dendrons) are typically not metric

measure spaces.

Given n sets, we call strings the elements of the direct product of the n sets.

A nonempty set of strings is well-connected if for every element v, there is another element v′, which differs from v only in its ith coordinate.

We expect to prove a conjecture of Wu and Xiong that every sufficiently large set of strings has a well-connected subset.

In a recent breakthrough, Adiprasito, Avvakumov, and Karasev constructed a triangulation of the n-dimensional real projective space with a sub-exponential number of vertices.

They reduced the problem to finding a small downward closed set-system F covering an n-element ground set which satisfies a certain condition.

We plan to study a variant of the problem, where the condition is strengthened.

In this case, we expect to prove that the size of the smallest F is strongly sub-exponential.

Thomassen formulated the following conjecture: Every 3-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree at most 1 and the red subgraph has minimum degree at least 1 and contains no 3-edge path.

Since all monochromatic components are small in this coloring and there is a certain irregularity, we call such a coloring crumby.

Recently, Bellitto, Klimošová, Merker, Witkowski and Yuditsky constructed an infinite family refuting the above conjecture.

Their prototype counterexample is 2-connected, planar, but contains a K_4-minor and also a 5-cycle.

This leaves the above conjecture open for some important graph classes: outerplanar graphs, K_4-minor-free graphs, bipartite graphs.

In this regard, we expect to prove that 2-connected outerplanar graphs, subdivisions of K_4 and 1-subdivisions of cubic graphs admit crumby colorings.

It is a fundamental extremal question to determine the maximum number of edges in a K_t-minor-free graph on n vertices.

Imposing a girth condition makes the graph sparser.

We expect to prove results for K_4, K_5-minor-free graphs of girth 5.

We expect to prove theorems related to geometric variants of the following classical graph-theoretic

problem of Erdős, Gallai, and Rogers. Given a K_r-free graph on n vertices and an integer s < r,

at least how many vertices can we find such that the subgraph induced by them is K_s-free?

In particular, we study string graphs.

We plan to construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number.

In the opposite direction, we expect to show that the class of disjointness graphs of (possibly self-intersecting) 2-way infinite polygonal chains of length 3 is χ-bounded.

Schmidt and Tuller posed a conjecture concerning optimal packing and covering of integers by translates of a given three-point set.

We expect to confirm their conjecture and relate it to several other problems in combinatorics.

The chromatic number of the plane is the smallest number of colours needed to colour it so that no two points at unit distance apart receive the same colour.

By a recent computer assisted break-through result of de Grey, the chromatic number of the plane is at least 5.

We plan to develop an approach that could lead to a human-verifiable proof of this fact.

Our ideas are based on finding almost monochromatic similar copies of certain sets in colourings of the plane and higher dimensional spaces.

We expect to prove several results about almost monochromatic sets that can be of independent interest.

A set is a k-distance set if its points span at most k different distances. Finding the maximum cardinalities of k-distance sets in d-dimensional space is a well-studied and difficult question, which is in general open.

We study the maximum cardinalities of nearly k-distance sets, which is an approximate version of k-distance sets.

Among other results, we expect to answer a question of Erdős, Makai and Pach proving that if the dimension is sufficiently large compared to k, then the two maximums are the same.

We also consider a related Turán type question about the maximum number of pairs among a given number of points whose distance is very close to k fixed distances.

A nonempty set of strings is well-connected if for every element v, there is another element v′, which differs from v only in its ith coordinate.

We expect to prove a conjecture of Wu and Xiong that every sufficiently large set of strings has a well-connected subset.

In a recent breakthrough, Adiprasito, Avvakumov, and Karasev constructed a triangulation of the n-dimensional real projective space with a sub-exponential number of vertices.

They reduced the problem to finding a small downward closed set-system F covering an n-element ground set which satisfies a certain condition.

We plan to study a variant of the problem, where the condition is strengthened.

In this case, we expect to prove that the size of the smallest F is strongly sub-exponential.

Thomassen formulated the following conjecture: Every 3-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree at most 1 and the red subgraph has minimum degree at least 1 and contains no 3-edge path.

Since all monochromatic components are small in this coloring and there is a certain irregularity, we call such a coloring crumby.

Recently, Bellitto, Klimošová, Merker, Witkowski and Yuditsky constructed an infinite family refuting the above conjecture.

Their prototype counterexample is 2-connected, planar, but contains a K_4-minor and also a 5-cycle.

This leaves the above conjecture open for some important graph classes: outerplanar graphs, K_4-minor-free graphs, bipartite graphs.

In this regard, we expect to prove that 2-connected outerplanar graphs, subdivisions of K_4 and 1-subdivisions of cubic graphs admit crumby colorings.

It is a fundamental extremal question to determine the maximum number of edges in a K_t-minor-free graph on n vertices.

Imposing a girth condition makes the graph sparser.

We expect to prove results for K_4, K_5-minor-free graphs of girth 5.

We expect to prove theorems related to geometric variants of the following classical graph-theoretic

problem of Erdős, Gallai, and Rogers. Given a K_r-free graph on n vertices and an integer s < r,

at least how many vertices can we find such that the subgraph induced by them is K_s-free?

In particular, we study string graphs.

We plan to construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number.

In the opposite direction, we expect to show that the class of disjointness graphs of (possibly self-intersecting) 2-way infinite polygonal chains of length 3 is χ-bounded.

Schmidt and Tuller posed a conjecture concerning optimal packing and covering of integers by translates of a given three-point set.

We expect to confirm their conjecture and relate it to several other problems in combinatorics.

The chromatic number of the plane is the smallest number of colours needed to colour it so that no two points at unit distance apart receive the same colour.

By a recent computer assisted break-through result of de Grey, the chromatic number of the plane is at least 5.

We plan to develop an approach that could lead to a human-verifiable proof of this fact.

Our ideas are based on finding almost monochromatic similar copies of certain sets in colourings of the plane and higher dimensional spaces.

We expect to prove several results about almost monochromatic sets that can be of independent interest.

A set is a k-distance set if its points span at most k different distances. Finding the maximum cardinalities of k-distance sets in d-dimensional space is a well-studied and difficult question, which is in general open.

We study the maximum cardinalities of nearly k-distance sets, which is an approximate version of k-distance sets.

Among other results, we expect to answer a question of Erdős, Makai and Pach proving that if the dimension is sufficiently large compared to k, then the two maximums are the same.

We also consider a related Turán type question about the maximum number of pairs among a given number of points whose distance is very close to k fixed distances.