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1 List of Problems Those question with a star next to them are considered slightly more challenging. Problems 9, 11, and 19 from the book The probabilistic method, by Alon and Spencer. Question 1. Prove the following inequalities, where k, n, αn N and n k for and. Do not use Stirling s Formula for and. For, recall H(α) = α log α (1 α) log(1 α). (n k) k ( ) n < nk ( k! ) k k! n < e nh(α) (d) αn ( n ) ( k n ( en ) k < ( k ) k) k 2n 4n as n. n πn Question 2. 1 Prove that for all n, N N with N n, R 2 (n) > N ( ) N 2 1 (n 2). n Deduce that R 2 (n) > e 1 n2 n/2 when n is large enough. Question 3. The Ramsey number R s (m, n) is the smallest number N such that in any red-blue colouring of [N] s, the complete system of s-element subsets of [N], there is a red [m] s or blue [n] s. Prove that R s (m, n) N whenever there exists p [0, 1] such that ( ) ( ) p (m s ) N + (1 p) s) N (n < 1 m n Does this give a polynomial or an exponential lower bound on R 3 (4, n) as a function of n? 2 Prove that R 3 (4, n) 2 cn for some constant c > 0. Question 4. Let Σ be a finite set. A word on Σ is an ordered list of symbols from Σ. A word is k-non-repetitive (k-nr) if no subword of length k appears at least twice. For each k N, Prove that every k-non-repetitive word has length at most Σ k + k 1. Prove that a random word of length N is k-nr with positive probability if N < (2 Σ ) k/2. Improve the bound in to N < ( Σ k 1)/(4ek) using the local lemma. (d) Find a determistic algorithm for constructing a k-nr word of length Σ k + k 1. Question 5 Prove that for each n N, there exists a set A [n] of size n/2 containing no arithmetic progression of length 2 log 2 n and such that no three consecutive integers in [n] are absent from A. For each ε > 0, prove that there exists a constant c(ε) > 0 and a set A [n] of size at most (1 + ε) n/2 such that A contains no arithmetic progression of length c(ε) log 2 n and no two consecutive integers in [n] are absent from A. Question 6. The independence number of a graph G, denoted α(g), is the maximum size of a set of vertices with no edges between them. 1 Hint : try deleting a vertex from each monochromatic copy of a complete graph K n of order n appearing in a random red-blue colouring. 2 Hint : Take a random orientation of the complete graph on n vertices, and then colour a triple red if it forms a directed triangle in the orientation, and blue otherwise. 1

2 Write down the expected number of independent sets of m vertices in G n,p. Show that for any ε > 0, P[α(G n,p ) > (2 + ε) log 1/(1 p) n] 0 as n. Question 7. The transversal number τ(h) of a hypergraph H is the minimum cardinality of a set T of elements of X which meet every member of H (i.e. every set in H contains at least one element from T ). Prove that if every element of H has size r (so H is a collection of r-element subsets of X), then for any p [0, 1], Deduce that τ(h) ( H + X log r)/r. τ(h) p X + (1 p) r H. Question 8. Let n 2 and let H be a hypergraph all of whose sets have size n. Prove that if H has 4 n 1 edges, then there is a colouring of the points in H with four colours so that none of the sets in H is monochromatic (a set is monochromatic if all of its elements receive the same colour in the colouring). Question 9. Prove that there exists a constant c > 1 such that in R d we can choose c d distinct points such that no three of the points form an angle of at least ninety degrees. 3 Question 10. Let S be a set of binary strings of length n 2. Then S is called universal if for each pair {i, j} {1, 2,..., n} and any binary string ab of length two, there is a string s S such that s i = a and s j = b. Prove that every universal set of strings of length n has size at least log 2 n. Prove that there is a universal set of binary strings of size at most 8 log n. Question 11. The list chromatic number χ l (G) of a graph G is the minimum number k such that if we assign to each vertex of the graph a list of colours of size k, it is possible to select a colour from each list to obtain a proper colouring of the graph. Show that every n-vertex bipartite graph G has χ l (G) log 2 n + 1. Show furthermore that there exist bipartite graphs G with χ l (G) log 2 n. Question 12. Prove that the proportion of subsets of [n] containing an arithmetic progression of length at least c log 2 n tends to zero if c > 2 and tends to one if c < 2, as n. Construct a set A [n] of size A n 1 1/k containing no arithmetic progression of length k + 1 using the method of alterations. Let A be the set of all integers less than n whose digits in base 2k 1 are all less than k. Show that A does not contain an arithmetic progression of length k + 1 and A n log 2k 1 k. Question 13. Let H be any graph with e > 1 edges and v vertices. Prove that there exists an n-vertex graph G not containing H such that e(g) n (2e v)/(e 1). Question 14. Consider the equation integer equation a 1 x 1 + a 2 x a k x k = 0 where a i Z\{0} and x i Z. Let a = {a i : a i > 0} and b = { a i : a i < 0}. Prove that if a b, then there is a subset of Z n of size n a+b with no solution (x 1, x 2,..., x k ) to the given integer equation. Using the example on sum-free sets in the notes, prove that if a b, then any set of positive integers of size n contains a subset of size n a+b with no solutions to the given integer equation. Question 15. A trivial solution to the Sidon equation x 1 + x 2 = x 3 + x 4 in an abelian group G, + is a solution (x 1, x 2, x 3, x 4 ) such that {x 1, x 2 } = {x 3, x 4 }. 3 Hint : pick points of the d-dimensional hypercube independently with probability p, and then consider the expected number of triples forming a right-angled triangle. 2

3 Prove that every finite set B Z contains a set A such that A B 1/3 and A has only trivial solutions to the Sidon equation. Let G = Z q, where q is prime. Suppose p be a prime such that q > 4p 2 and let x p denote the reduction of an integer x modulo p. Prove that the set A = {x + 4px 2 p : 0 x < p} has only trivial solutions to the Sidon equation in G. Prove that the following set is a Sidon set: A n = { 6n 2 log p : p is prime, p n}. You may which to observe log x log y 1/2y when x, y > 0 are integers. Determine log A n lim n log n. (d) Improve the result of to A B 1/2. Question 16. Let n N. A maximal chain in [n] is a sequence of subsets (A i ) n i=0 with A i A i+1 and A i = i. An antichain in [n] is a family of sets where no set is contained in another. How many maximal chains in [n] contain a fixed subset of [n] of size r? Let A be an antichain of subsets of [n]. Show, by taking a randomly and uniformly chosen maximal chain in [n], that n A r ( n 1 r) where A r = {A A : A = r}. r=0 Use to show that A ( n n/2 ). Question 17. A generalized triangle comprises three sets A 1, A 2, A 3 with A 1 A 2 A 3 = and with A 1 A 2, A 2 A 3, A 3 A 1 all non-empty. The family of all s-element subsets of [n] is denoted [n] s. For n 5, find a family A [n] 3 of size ( ) n 1 2 containing no generalized triangle. Let A [n] 3 contain no generalized triangle. For x, y [n], let (x, y) = {A A : {x, y} A} and let X xy be a Bernouilli random variable with success probability ( n 2) 1 (x, y) 1. For A A let X A = {x,y} A X xy. Show that if n 5, then E[X A ] Let Y = A A X A. Show that E[Y ] 1. (d) Combine and to obtain A ( ) n (n 1)(n 2) Question 18. Let A be a finite set of real numbers, and define A + A = {a + a : a, a A} and A A = {a a : a, a A}. Using the appropriate results from the notes, prove that A + A A A A 5/2. Give an explicit example of a set A with A + A A A A 4. Give an example of a set A [ A 2 ] with A + A A A A 4. You may assume the unbelievably strong statement τ(i) i/(log i) 2 for every even integer i, where τ(i) is the number of ways of writing i as a sum of two primes. 3

4 Question 19. Prove that P[X = 0] σ2 µ 2 when X has mean µ 0 and variance σ 2. Then show that the stronger inequality below is true: P[X = 0] σ2 µ 2 + σ 2. Let X and Y be discrete i.i.d real-valued random variables. Prove that Question 20. P[ X Y 2] 3P[ X Y 1]. Show that the three cannot be replaced with a smaller number. Recall the definition of crossing number cr(g) of a graph G from the notes. A multigraph is a pair G = (V, E) where V is a set of vertices and E is a multiset of pairs of elements of V (in other words, the same as a graph except we allow two vertices to be joined by many edges). Prove that every element of E appears at most k times in E, then cr(g) + k 2 V E 3 k V 2. Let P be a set of points in the plane. Let d(p ) denote the set of distinct distances between points of P, namely {d(u, v) : u, v P }. Prove that d(p ) P 2/3. You might wish to consider concentric circles around each p P which cover all the other points of P, and consider the graph formed by shortest arcs between vertices of P of all of these circles. Then apply. Use the Szemerédi-Trotter Theorem to improve the bound in to P 4/5. Question 21. Let d N. A curve in R 2 has degree d if it is the graph of y = P (x) for some polynomial P of degree d. Prove that up to a absolute constant factor depending on d, the Szemerédi-Trotter Theorem holds for incidences between a set of n points and m curves of degree d in R 2. Prove that the maximum number U(n) of unit distances determined by n distinct points in the plane satisfies U(n) n 3/2. Use to prove U(n) n 4/3. (d) Via a construction, give the best lower bound you can find on U(n). 4

5 Question 22. Let X 1, X 2,..., X n be independent random variables, and let X be their sum. Check that if var[x] = σ 2, then n σ 2 = var[x i ]. Let Y be a random variable with Y 1 and E(Y ) = 0. Prove that for 0 t 1, i=1 E(e ty ) 1 + t 2 var(y ). Apply parts and to show that if X 1, X 2,..., X n and X are as in and X i 1 and E[X i ] = 0 for all i, then for 0 t 1, E(e tx ) e t2 σ 2. (d) Use Markov s Inequality to prove that P( X λσ) 2e λ2 /4 where X is as in and 0 λ 2σ. Question 23. A random walk in the plane is walk on the lattice Z Z in which the direction of each step is chosen uniformly from all four possible directions, independently for different steps. Prove that the probability that the random walk is at distance at least λ n from the origin in n steps is at most 8e λ2 /4 for any positive real λ. Determine the probability that the random walk returns to the origin infinitely often. Question 24. Let X = [n], let S be a family of subsets of X, and let χ be a map X { 1, 1}. The discrepancy of χ is { } disc(χ, S) := max χ(x) : E S. The discrepancy of S is defined by disc(s) := min{disc(χ, S) : χ { 1, 1} X }. Let S consist of all k-term progressions in X. Show disc(s) k log n. Show that if S = n, then disc(s) n log n. Prove that there is a set system S on X with S = n and disc(s) n. Question 25. Let k be a positive integer. A k-covering of K n is a collection of bipartite subgraphs of K n for which every edge of K n appears in exactly k of the bipartite graphs. Let f k (n) denote the smallest number of bipartite graphs in a k-covering of K n. Prove that f 1 (n) = log 2 n and f k (n) log 2 n. x E Show that f k (n) log n + k, where the implied constant is absolute. Let φ(p ) denote the largest angle formed by any three points of the set P R 2, where three collinear points are considered to form an angle of π radians. Let φ(n) be the smallest value of φ(p ) over all n-point sets P R 2. Find φ(n) for n {3, 4, 5}. Prove that π φ(n) π log 2 n as n. [Hint : For a lower bound on φ(n), apply as follows: let P be a set of n points and define H r to be the graph of all pairs {p, q} P spanning a line with gradient tan θ, where (r 1)π θn < rπ. Show that each H r is bipartite and that the H r form a 1-covering of K n. 5

6 Question 26. Let γ(g) be the size of a smallest dominating set in a graph G. Find a sequence G n,d of n-vertex graphs of minimum degree at least d for which γ(g n,d ) n log d d. Question 27. Let X be a finite set, X 2 := X X. For a function f : X 2 { 1, 1}, define f : X 3 { 1, 1} and f : X 4 { 1, 1} by f (x 1, x 2, x 3 ) = f(x 1, x 2 )f(x 1, x 3 )f(x 2, x 3 ) f (x 1, x 2, x 3, x 4 ) = f(x 1, x 2 )f(x 3, x 2 )f(x 3, x 4 )f(x 1, x 4 ). Find an f : X 2 { 1, 1} such that f (x) x X 3 {0, 1}. Prove that almost all functions f : X 2 { 1, 1} with X = n satisfy f (x) n 3. x X 4 Find explicit functions f n : [n] { 1, 1} satisfying the inequality in. Question 28. Let n and be positive integers. Design a probability space and events A 1, A 2,..., A n therein such that P(A i ) = 1 +1, a dependency graph for the A i has maximum degree, and P ( n ) A i = 0. i=1 Prove the modified version of the local lemma, by slightly modifying the proof of the local lemma given in the notes. Question 29. Let G be a graph with no multiple edges, and let N(G) be the smallest positive integer N such that it is possible to associate an element x(v) Z N with vertex v V (G) so all elements of {x(u) + x(v) mod N : {u, v} E(G)} are distinct. Prove that N(G) E(G) for every non-empty graph G. For each positive integer m, find a graph G with m edges such that N(G) = m. Prove that for any graph G of maximum degree, N(G) 2 V (G). You may wish to randomly choose a number in Z N for each vertex, and apply the local lemma to the events A ef that the ends of e, f E(G) have the same sum modulo N. Question 30. Let G be a d-regular graph where d 4k 3. Prove that there is a partition (V 1, V 2,..., V k ) of V (G) such that for every vertex v V (G) and 1 i k, e(v, V i )) d ( d log d < 4 k k where e(v, V i ) is the number of edges of the form {v, w} with w V i. ) 1 2 6

7 Question 31. Let S be a finite set of integers. A vertex-distinguishing S-colouring of a graph G is a map χ : E(G) S such that for every edge {u, v} E(G), χ(f). e u χ(e) f v Prove, for all n 3, that the complete graph K n has a vertex-distinguishing S-colouring for some set S with S = 3. Prove that there exists a finite set S such that if d is large enough, then every d-regular graph has a vertex-distinguishing S-colouring. Question 32. Let K n,n denote the complete bipartite graph with n vertices in each part. Let χ be an edgecolouring of K n,n such that each colour appears at most 1 64 (n 4) times. Prove that for any χ, there is a perfect matching in K n,n all of whose edges have different colours. Show that if n is large enough, then in any proper colouring of K n,n, there is a perfect matching which does not contain four edges of the same colour. Deduce that in a latin square L, we can find a sequence of n entries, no two in the same row or column, such that no four of the entries are the same. Question 33. Let d N, c R +, and let G n be an n-vertex d-regular graph. Suppose that vertices of G n are selected independently with probability p, where p c(dn) 1/2 as n. Let X n be the number of edges between selected vertices of G n. Prove that P(X n = 0) e 1 2 c2 as n. Question 34. * Let p c (Z d ) denote the critical threshold for percolation in the d-dimensional integer lattice. Prove that 0 < p c (Z d ) < 1. Find a cubic lattice L a lattice where every element is immediately comparable to exactly three others such that p c (L) = 0. Suppose we randomly select vertices of a finite cubic graph independently with probability p, and at any time, a vertex becomes selected if it has at least two selected neighbors. Show that if p < 1 2, the probability that all vertices are selected within finite time is o(1). Question 35. Let H denote a random k-uniform hypergraph on [n], in which hyperedges appear uniformly and independently. Let X denote the number of isolated vertices of H. Prove that a sharp threshold function for the event X = 0 is given by τ(n) = log n ). ( n 1 k 1 Determine a threshold τ(n) for a k-term arithmetic progression in a subset of [n] whose elements are chosen uniformly and independently. Determine asymptotically the probability of a k-term progression when elements are chosen with probability c τ(n), where c R +. 7

8 Question 36. Let G be a graph whose vertices are independently infected at time zero with probability p. Suppose that any vertex v becomes infected if at least half its neighbours are infected. Let A = A G be the event that the entire graph becomes infected in finite time. Show that if G has n vertices and minimum degree at least ω log n, where ω = ω(n), then P(A) tends to zero or one according as p < 1 2 or p > 1 2. Give examples where P(A) 0 when p = 1 2 and where P(A) 1 when p = 1 2. Prove that if G has no cycles of length at most four, then P(A) 0 when p = 1 2. If G is the n n torus, show that no configuration of at most n 1 infected sites is in A. Prove that for the n n square grid, if pn/ log n, then P(A) 1. Question 37. Prove that if Z : Ω [0, 1] is a random variable and f : [0, 1] R is a convex function, then E(f(Z)) E(Z)f(1) + (1 E(Z))f(0). Let X 1, X 2,..., X n be negatively correlated random variables with expectation µ and range [N], and S = n i=1 X i. Prove that P(S E(S) λ) e λ2 2µnN. Question 38. Let ɛ R +. Let X denote the number of triangles in G n,p and let µ = p 3( n 3). Suppose that p 2 n 0. Prove that for some constant a > 0, P(X (1 ɛ)µ) e aɛ2 p 3 n 3. Prove that for some constant b > 0, P(X (1 + ɛ)µ) e bɛ2 p 3 n 2. Question 39. Let (X i ) i N be a martingale with difference sequence (Y i ) i N and Y 1 = X 1. Then (X i ) i N is c- Lipschitz with exceptional probability η if for all i N where i > 1, P( Y i > c) η. Prove that for λ 0, P( X n E(X n ) > λ) 2e λ2 /8nc 2 + 2nη. Let G be a graph of maximum degree. Colour the vertices of G independently with a uniformly chosen colour from [k] where k N. Uncolour any two adjacent vertices of the same colour. Show that if > log n and X is the number of vertices which retain their colour, then P( X E(X) > ω(n)(n log n) 1/2 ) 0 for any function ω(n). 8

9 Question 40. Let X 1, X 2,..., X n be random variables where X i : Ω R, and let Y i be measurable with respect to the σ-field F i generated by X 1, X 2,..., X i. Suppose that for some constants a i, c i R, E(Y i Y i 1 F i 1 ) < a i and Y i Y i 1 a i < c i a.s. Let A := n i=1 a i. Show that for λ > 0, ) P(max Y i > Y 0 + A + λ) < exp ( λ2 i [n] 2 c 2. i Let G 0 be the empty graph on n vertices, and let G i be formed from G i 1 for i N by adding an edge between a uniformly chosen pair of non-adjacent vertices of degree at most two, if such a pair exists, otherwise let G i = G i 1. Prove that for any ε > 0, if t (1 + ε)n, then G t a.a.s contains a component with a linear number of vertices as n. Question 41. Let X : Ω R + be an f(s) = ds certifiable k-lipschitz random variable, where Ω = r i=1 Ω i. Prove that E(X) M(X) kγ de(x) for some constant γ, where M(X) is a median of X. You might recall E(X) 1 2M(X) and apply Talagrand to each of the quantities P( X M(X) > ik dm(x)) for i 0. Let k 2, and let H be a k-uniform hypergraph. Let H be a random subgraph of H consisting of edges of H chosen independently with probability p. Prove that {E : E H } is highly concentrated at its expectation when H. Question 42. (d) Throughout this question, G is a triangle-free graph of maximum degree. Let χ(g) denote the chromatic number of G. Prove that if there is a proper ( + 1 r)-colouring of a subset of vertices of G such that at least r colours appear at least twice in the neighbourhood of every vertex of G, then χ(g) + 1 r. Prove that G is contained in a -regular triangle-free graph. Consider the following colouring procedure on a -regular graph: assign to each vertex independently and uniformly a colour from [d] where d = /2, and remove the colour on any vertex which has the same colour as one of its neighbours. Let X v denote the number of colours appearing at least twice in the neighbourhood of v. Prove that E(X v ) /e 6 1. Use Talagrand s Inequality to show that P( X v E(X v ) > (log ) E(X v ) < (e) Use the local lemma to prove that if (G) = is large enough, then with positive probability, none of the events X v < /2e occur. Deduce from and that χ(g) (1 1/2e 6 ) whenever G is a triangle-free graph of maximum degree. Question 43. Let B t (n) denote a random n n bipartite graph formed by adding edges t times to the empty n n bipartite graph t times, where a new edge is chosen uniformly from the set of pairs of vertices which are not already edges. Prove that the random variables M = min{t : B t (n) has a perfect matching} and min{t : B t (n) has no isolated vertices} are a.a.s equal, and determine an asymptotic formula for M as a function of n. 9

10 Question 44. Let V be an n-element set and k an integer such that kn is even. Let f be a uniformly chosen pairing of V [k], and let G(k) denote the multigraph on V obtained by contracting all vertices (v, i) : 1 i k to a single vertex v V. Let A be the event that G(k) has no multiple edges and no loops. Using Poisson approximation, show that for each fixed k, P(A) e 1 4 (1 k2 ) Deduce from the asymptotic number of k-regular simple graphs (no multiple edges or loops allowed) on n vertices. 10