Topics in metric geometry, combinatorial geometry, extremal combinatorics and additive ...

235 Pages·2017·0.97 MB·English

Topics in metric geometry, combinatorial
geometry, extremal combinatorics and
additive combinatorics
Luka Mili(cid:19)cevi(cid:19)c
Trinity College
and
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
June, 2017

Topics in metric geometry, combinatorial geometry,
extremal combinatorics and additive combinatorics
Luka Milic(cid:19)evic(cid:19)
Abstract
In this thesis, we consider several combinatorial topics, belonging to the areas
appearing in the thesis title.
Given a non-empty complete metric space (X;d), a family of n continuous
maps f ;f ;:::;f :X ! X is a contractive family if there exists (cid:21) < 1 such that
1 2 n
for any x;y 2 X we have d(f (x);f (y)) (cid:20) (cid:21)d(x;y) for some i. In the (cid:12)rst part
i i
of the thesis, we
(i) construct a compact metric space (X;d) with a contractive family ff;gg,
such that no word in f;g has a (cid:12)xed point, and
(ii) show that if ff;g;hg is a contractive family such that f;g;h commute and
(cid:21) < 10(cid:0)23, then they have a common (cid:12)xed point.
The proofs of these two statements are combinatorial in nature. For (i), we
introduce a new concept of a diameter space, leading us naturally to a combi-
natorial problem about constructing certain sets of words. The result (ii) has a
Ramsey-theoretic(cid:13)avour, andisbasedonstudyingthelocalandglobalstructure
of a related metric space on N3. These answer questions of Austin and Stein.
In the second part, we prove that given any 4-colouring of the edges of K ,
n
we can (cid:12)nd sets X;Y;Z and colours x;y;z (not necessarily distinct) such that
X [ Y [ Z = V(K ), and each of K [X;x];K [Y;y] and K [Z;z] has diame-
n n n n
ter bounded by 160 (where K [X;x] denotes the edges in X that have colour
N
x). This theorem is motivated by the work on commuting contractive families,
where the analogous statement for 3 colours played a crucial role, and by the
Lov(cid:19)asz-Ryser conjecture. The proof is in the spirit of structural graph theory.
The key point is the fact that the diameters are bounded. This strengthens a
result of Gy(cid:19)arfa(cid:19)s, who proved the same but with no diameter bounds (i.e. just
with the sets being connected).
Recall that a set of points in Rd is in general position if no d + 1 lie on a
common hyperplane. Similarly, we say that a set of points in Rd is in almost
general position if no d+2 lie on a common hyperplane. In the third part, we
answer a question of Fu(cid:127)redi, by showing that, for each d, there are sets of n
points in almost general position in Rd, whose subsets in general position have
size at most o(n). The proof is based on algebraically studying to what extent
polynomial maps preserve cohyperplanarity, and an application of the density
version of the Hales{Jewett theorem.
In the fourth part, we answer a question of Nathanson in additive combina-
torics about sums, differences and products of sets in Z (the integers modulo
N
N). For all ϵ > 0 and k 2 N, we construct a subset A (cid:26) Z for some N, such
N
that jA2 +kAj(cid:20) ϵN, while A(cid:0)A = Z . (Here A(cid:0)A = fa (cid:0)a : a ;a 2 Ag
N 1 2 1 2
and A2 +kA = fa a +a′ +a′ +(cid:1)(cid:1)(cid:1)+a′ : a ;a ;a′;a′;:::;a′ 2 Ag.) We also
1 2 1 2 k 1 2 1 2 k
prove some extensions of this result. Among other ingredients, the proof also
includes an application of a quantitative equidistribution result for polynomials.
In the (cid:12)nal part, we consider the Graham-Pollak problem for hypergraphs.
Let f (n) be the minimum number of complete r-partite r-graphs needed to
r
partition the edge set of the complete r-uniform hypergraph on n vertices. We
( )
disprove a conjecture that f (n) (cid:21) (1 + o(1)) n , by showing that f (n) (cid:20)
( ) 4 2 4
14(1 + o(1)) n . The proof is based on the relationship between this problem
15 2
and a problem about decomposing products of complete graphs, and under-
standing how the Graham-Pollak theorem (for graphs) affects what can happen
here.

Declaration
This dissertation is the result of my own work and includes nothing which is
theoutcomeofworkdoneincollaborationexceptforChapter5, whichrepresents
joint work with Imre Leader and Ta Sheng Tan. It is not substantially the same
as any that I have submitted, or, is being concurrently submitted for a degree
or diploma or other quali(cid:12)cation at the University of Cambridge or any other
University or similar institution. I further state that no substantial part of my
dissertation has already been submitted, or, is being concurrently submitted for
any such degree, diploma or other quali(cid:12)cation at the University of Cambridge
or any other University or similar institution. It does not exceed the prescribed
word limit for the relevant Degree Committee.

Acknowledgements
I am very grateful to my supervisor, Prof Imre Leader, for all the stimulating
mathematical discussions and for the helpful advice. He was a truly excellent
and wise supervisor, and most importantly, he has always had understanding for
me and encouraged me to pursue my own intuition.
I would like to thank Vytautas Gruslys, for all the brilliant mini-seminars he
prepared. Some of the most exciting mathematics that I know, I learned from
him.
I am grateful to Dr Ta Sheng Tan, for collaborating with me and for helping
me with preparing some of the text for this thesis.
I would also like to thank Prof Timothy Gowers, for the work we did together
in the last stage of my PhD (these results do not appear in the thesis). Working
with him has been a very rewarding experience.
My thanks also go to Trinity College and Department of Pure Mathematics
and Mathematical Statistics for their generous (cid:12)nancial support and hospitality.
Finally, I am indebted to my wife Teodora, for all her love, support, encour-
agement and patience. Over the past few years, she helped me realize that I
should not be obsessed with the results, but that I should remember instead to
enjoy the work as much as possible, and that the beauty will emerge in mathe-
matics, once the mathematical results become a labour of love. She has always
inspired me to pose questions and to stay curious. The things that I learned
from her are the things that I treasure the most in my PhD studies. I dedicate
this thesis to her.