About my eating habits: I usually go to a nearby restaurant to eat. Sometimes I bring away the food and eat it in the company kitchen, but sometimes I go with a group of random people there.

Background: I am female, my colleague is male. Cultural info: In Hungary lunch is the major meal of the day, not dinner.

A new colleague has come to our company. My company has a kitchen where most of the people eat (there are huge tables, micros, fridge, etc). Those who don't want to eat in the kitchen, can choose to sit in some restaurants around, or eat at their tables. So there are quite a lot of possibilities. The new colleague of mine, instead of eating in the kitchen, connect to an eating group or whatever, continuously writes me on a private chat and asks me to eat with him in one of the nearby restaurants.

The first time it happened, he didn't ask me on chat, but when he got to know that I am holding a lunch break in a nearby restaurant, he stated that he is going with me. I got afraid and quickly called another colleague to come with us so that I am not alone with him. The new colleague was surprised and remained quite quiet during the lunch.

Afterwards, it got worse. He continued asking me in a private chat window to eat together, instead of asking me in the presence of others. I got afraid and said no with some poor excuse. This continued almost every day.

I don't want to be rude to him, but for me it would be very uncomfortable to always eat with a colleague privately in a restaurant. I wanted first to introduce him to other restaurant visitors or something, but since he is very stubborn and continues asking me every day in a private chat, I started to be afraid of the possibility of eating with him - even in others' presence - and couldn't do that. Maybe I am too sensitive and shy, but it always freaks me out if somebody is too pushy. Every normal person would understand after 3-4 'no' and there are plenty other colleagues he can ask to eat with in restaurants, and the kitchen is always an option to meet many. He is not that 'helpless'.

I asked my family for advice and they said to go with him eating and bring others as well. I feel that maybe this would be a solution, but just the thought itself makes me nervous after the daily ping I receive from him. I would not mind if he would eat in the kitchen and I would be there also among others. But to go out to a restaurant with the possibility that I cannot find every day some volunteers to eat with us is not an acceptable option for me. I don't want to hunt people every day to play the 'bodyguard' for me.

My question, how can I effectively avoid eating with him in a restaurant and not make him turn to be an enemy?

Update: I got a new invitation today, quite lately. This time I was prepared, and tried with a polite but clear answer, without extra excuses:
"Thank you for asking me out for lunch again, but I really don't want to."
He just answered with an "I see" and an "Ok". I expected the question "why" but it didn't come. If there will be following cases, I will honestly say to not ask me again.

Update 2.

So, I decided to keep many of the advices here ready to fire and wait a couple of days to see the result.

Summary:

• Following @Mr Positive's suggestion, I try to eat with other colleagues, friends of mine every day.
• Since the last chat interaction I didn't get any new chat message.
• The colleague in question developed a new habit a day after: he walked up to my table and hold a long (one-way) speech about work/workplace related stuff. He repeated this action later as well. From my part, I remained passive, and the second time I cut him down telling that I have to concentrate to my work (which is the truth).
• On Friday while I was about to depart to eat he came to speak again, and - as if suddenly realizing the time - he asked "oh, sorry, I take your lunch time. [Restaurant Name]?" I took @Kate Gregory's answer to say to stop it. It was just extra lucky that other colleagues heard it too. To his upcoming question "But you frequently go to that restaurant, don't you?" I didn't answer but repeated what I said like a parrot.
• Today is Monday. Lunch time he stopped nearby my table and asked the "air" if anyone is going to eat and left since there was no answer. Then 2 mins later he came back, so he was not eating. I didn't see him going to eat but I went out after a while so I don't know. He didn't tried to walk up to my table till now either. I think this is a progress.
  I was refusing to take the advice to talk to any manager first, but then the occasion today at lunch time made me finally do this step also. One of our main boss was in the group I ate with. I asked to speak with him after lunch and I told the whole story as it is, and added that I see the progress today and hope this is a good sign that he will leave me in peace from now on. We agreed that I should report if he doesn't change permanently and in this case the boss will make steps.

Thank you for all the advices and that you cared to reply. With your help it seems I managed to avoid to lunch with him at all. Since I made myself clear and reported the issue as well I think I don't have to worry about any improper actions from his part now on.